2.1 A priori \(L^{2}\) estimates
We start with the basic energy identity.
Lemma 1
For time and space variables \((t, x)\in [0,T]\times [0,l]\), the conservation of mass and energy is given as
$$\begin{aligned}& \int _{0}^{l}\rho (y,t)\,dy= \int _{0}^{l}\rho (y, 0)\,dy, \end{aligned}$$
(2.1)
$$\begin{aligned}& \int _{0}^{l} \biggl(c_{v}\rho _{0}\theta +\frac{1}{2} \vert \rho _{0}{{u}} \vert ^{2} + \frac{1}{2g} \vert {d_{y}} \vert ^{2} \biggr) (y,t)\,dy = E_{0}, \end{aligned}$$
(2.2)
where
$$ E_{0}= \int _{0}^{l} \biggl(c_{v}\rho _{0}\theta +\frac{1}{2}\rho _{0} \vert {u} \vert ^{2} + \frac{1}{2g} \vert {d_{y}} \vert ^{2} \biggr) (y,0)\,dy. $$
Proof
Integrating (1.8)1 with respect to space and time and using boundary condition (1.10), it is easy to see that equality (2.1) holds.
In order to prove (2.2), we multiply (1.8)2 by u. The resulting equation is then integrated over [0,l]; after integration by parts, we obtain
$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \int _{0}^{l}\rho _{0}u^{2}\,dy+ \int _{0}^{l} \frac{u_{y}^{2}}{g}\,dy= \int _{0}^{l} \biggl(p+\frac{1}{2g^{2}} \vert d_{y} \vert ^{2} \biggr)u_{y}\,dy. \end{aligned}$$
(2.3)
On the other hand, taking dot product of equation (1.8)3 with \(- (\frac{1}{g} (\frac{d_{y}}{g} )_{y} + \frac{\vert {d}_{y}\vert ^{2}d}{g^{2}} )\), as a result one can get
$$\begin{aligned} - d_{t} \cdot \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} + \frac{ \vert {d}_{y} \vert ^{2}}{g^{2}}d\biggr) = - \biggl\vert \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} + \frac{1}{g^{2}} \vert d_{y} \vert ^{2}d \biggr\vert ^{2}. \end{aligned}$$
(2.4)
The left-hand side of (2.4) can be written as
$$\begin{aligned} - d_{t} \cdot \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} + \frac{ \vert {d}_{y} \vert ^{2}}{g^{2}} d \biggr)= -d_{t}\cdot \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} - d_{t}\cdot \frac{ \vert {d}_{y} \vert ^{2}}{g^{2}} d= -d_{t}\cdot \frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y}, \end{aligned}$$
(2.5)
where we have used constraint \(\vert d\vert ^{2}=1\). Plugging (2.5) into (2.4) and then multiplying the resulting equation by g, we get
$$\begin{aligned} -d_{t}\cdot \biggl(\frac{d_{y}}{g} \biggr)_{y} =- g \biggl\vert \frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} + \frac{1}{g^{2}} \vert d_{y} \vert ^{2}d \biggr\vert ^{2}. \end{aligned}$$
(2.6)
Then, we integrate equality (2.6) over [0,l], and we have
$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \int _{0}^{l}\frac{ \vert d_{y} \vert ^{2}}{g}\,dy+ \int _{0}^{l} g \biggl\vert \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y}+ \frac{1}{g^{2}} \vert d_{y} \vert ^{2}d \biggr\vert ^{2}\,dy=- \int _{0}^{l} \frac{1}{2g^{2}} \vert d_{y} \vert ^{2} u_{y}\,dy. \end{aligned}$$
(2.7)
Combining (2.3) and (2.7) with integral form of the temperature equation (1.8)4 implies equality (2.2). □
Next, we carry out the estimate on the lower bound of g. To this end, we perform some calculations in the spirit of [35] as preparations. Now, we integrate momentum conservation equation with respect to time t, and using first equation of (1.8), we can get
$$ \rho _{0}u(y,t) - \rho _{0}u_{0}(y) + \int _{0}^{t} \biggl(p + \frac{1}{2g^{2}} \vert d_{y} \vert ^{2} \biggr)_{y}(y, s)\,ds = \frac{d}{dy} \bigl(\ln g(y,t) - \ln g_{0}(y) \bigr). $$
(2.8)
Integrating (2.8) with respect to y from the point \(\xi (t)\) to an arbitrary point y for any fixed time t, we get
$$\begin{aligned} & \int _{\xi (t)}^{y}\bigl(\rho _{0}u(y,t) - \rho _{0}u_{0}(y)\bigr)\,dy + \int _{0}^{t} \biggl(p + \frac{1}{2g^{2}} \vert d_{y} \vert ^{2} \biggr) (y, s)\,ds \\ &\quad {} - \int _{0}^{t} \biggl(p + \frac{1}{2g_{0}^{2}} \vert d_{y} \vert ^{2} \biggr) \bigl(\xi (t), s\bigr)\,ds = \ln g(y,t) - \ln g_{0}\bigl(\xi (t),t\bigr). \end{aligned}$$
(2.9)
After rewriting, we have
$$ \begin{gathered} \exp \biggl( \int _{0}^{t} p \,ds \biggr)= g(y,t)X(t)Y(y,t)Z(y,t), \end{gathered} $$
(2.10)
where
$$\begin{aligned}& X(t)= \frac{1}{g(\xi (t),t)}\exp \biggl( \int _{0}^{t}p+ \frac{1}{2g^{2}} \vert d_{y} \vert ^{2}\bigl(\xi (t), s\bigr)\,ds \biggr), \\& Y(y,t) = \exp \biggl( \int _{\xi (t)}^{y} \rho _{0}u_{0}(x)- \rho _{0}u(t,x)\,dx \biggr), \\& Z(y,t)= \exp \biggl(- \int _{0}^{t}\frac{1}{2g^{2}} \vert d_{y} \vert ^{2}\,ds \biggr). \end{aligned}$$
Multiplying equation (2.10) by \(R\rho _{0}\theta \) and integrating the result over \([0,t]\), we have
$$ \exp \biggl( \int _{0}^{t} p \,ds \biggr)-1=R\rho _{0} \int _{0}^{t}X(s)Y(y,s)Z(y,s) \theta \,ds. $$
(2.11)
From (2.10) and (2.11) we get
$$ g(y,t)= X(t)^{-1}Y(y,t)^{-1}Z(y,t)^{-1} \biggl(1 + R \int _{0}^{t} X(s)Y(y,s)Z(y,s) \rho _{0}(y)\theta (y,s)\,ds \biggr). $$
(2.12)
A prior positive lower and upper bound of g is stated in the following proposition.
Proposition 1
Given \(T\in (0,\infty )\), it holds that
$$\begin{aligned}& g \geq C, \\& \Vert g \Vert _{\infty }(t) \leq C \bigl\Vert Z^{-1} \bigr\Vert _{\infty } \biggl(1+\frac{R}{\mu } \int _{0}^{t} \Vert \rho _{0}\theta \Vert _{\infty }\,d\tau \biggr) \end{aligned}$$
for any \(t\in [0,\infty )\).
Proof
By Lemma 1, it follows from the Hölder inequality that
$$\begin{aligned} \biggl\vert \int _{0}^{l}\rho _{0}(u-u_{0})\,d\xi \biggr\vert \leq & \biggl( \int \rho _{0}\,d\xi \biggr)^{\frac{1}{2}} \biggl[ \biggl( \int _{0}^{l} \rho _{0}u^{2}\,d\xi \biggr)^{\frac{1}{2}}+ \biggl( \int _{0}^{l}\rho _{0}u_{0}^{2}\,d\xi \biggr)^{\frac{1}{2}} \biggr] \\ \leq &2\sqrt{2 \Vert \rho _{0} \Vert _{1} E_{0}}. \end{aligned}$$
Therefore, it follows from the definition of Y in Lemma 1 that
$$\begin{aligned}& \exp \bigl\{ -C\sqrt{ \Vert \rho _{0} \Vert _{1} E_{0}} \bigr\} \leq Y(y,t) \leq \exp \bigl\{ C\sqrt{ \Vert \rho _{0} \Vert _{1} E_{0}} \bigr\} , \\& C^{-1}\leq Y(y,t) \leq C. \end{aligned}$$
Similarly,
$$ 0< Z = e^{\int _{0}^{t}-\frac{\vert d_{y}\vert ^{2}}{g^{2}}} < 1. $$
(2.13)
Now we show that \(X(t)\) is bounded above and below by the initial data. Assume that
$$ \mathcal{F}(y,t)= \int _{0}^{t} \biggl(\frac{u_{y}}{g}-p- \frac{1}{2g^{2}} \vert {d_{y}} \vert ^{2} \biggr) (y,s)\,ds+ \int _{0}^{y}\rho _{0} u_{0}(x)\,dx, $$
(2.14)
then
$$ \mathcal{F}_{y}= \rho _{0}u,\qquad \mathcal{F}_{t}= \frac{u_{y}}{g}-p- \frac{1}{2g^{2}} \vert d_{y} \vert ^{2}. $$
With the aid of the continuity equation, we obtain
$$ (g\mathcal{F})_{t} - (u\mathcal{F})_{y}= u_{y} - \rho _{0}\theta - \frac{1}{2g} \vert d_{y} \vert ^{2}- \rho _{0}u^{2}. $$
Integrating it over \([0,l]\times [0,t]\), using non-slip boundary (1.10) condition, we get
$$ \int _{0}^{l}g\mathcal{F}\,dy = \int _{0}^{l}g_{0}\mathcal{F}_{0}\,dy- \int _{0}^{t} \int _{0}^{l} \biggl(\rho _{0}\theta + \frac{1}{2g} \vert d_{y} \vert ^{2}+ \rho _{0}u^{2} \biggr)\,dy\,ds. $$
By using mass conservation (2.1), we have
$$ \int _{0}^{l}g\,dy= \int _{0}^{l}g_{0}\,dy. $$
Using the continuity of g, for \(t>0\), there exists \(\xi (t)\in [0,l]\) such that
$$ \mathcal{F}\bigl(\xi (t), t\bigr)= \frac{1}{g_{0}} \int _{0}^{l}g\mathcal{F}(y,t)\,dy. $$
On the other hand, since \(u_{y}=g_{t}\), then from the definition of \(\mathcal{F}\) we have
$$ \mathcal{F}\bigl(\xi (t),t\bigr)=\ln g\bigl(\xi (t),t\bigr)- \int _{0}^{t} \biggl(p+ \frac{1}{2g^{2}} \vert {d_{y}} \vert ^{2} \biggr) \bigl(\xi (t),s\bigr)\,ds+ \int _{0}^{\xi (t)} \rho _{0} u_{0}(x)\,dx. $$
(2.15)
By rewriting in the form
$$\begin{aligned} &\frac{1}{g(\xi (t), t)}\exp \biggl( \int _{0}^{t} \biggl(\rho _{0} \theta + \frac{1}{2g^{2}} \vert d_{y} \vert ^{2} \biggr) \bigl( \xi (t), s\bigr)\,ds \biggr) \\ &\quad =\exp \biggl(\frac{1}{g_{0}} \int _{0}^{t} \int _{0}^{l} \biggl(\rho _{0} \theta + \frac{1}{2g} \vert d_{y} \vert ^{2}+ \rho _{0}u^{2} \biggr)\,dy\,ds + \int _{0}^{a(t)} \rho _{0}u\bigl(\xi (t),t \bigr)\,dx \biggr). \end{aligned}$$
(2.16)
Thus it is not hard to see that
$$ C^{-1}\leq X(t)\leq C. $$
(2.17)
Recalling \(p\geq 0\), we obtain
$$\begin{aligned} g =&X^{-1}Y^{-1}Z^{-1}e^{\frac{1}{\mu }\int _{0}^{t}p \,d\tau }, \\ \geq &C, \end{aligned}$$
the conclusion follows. □
Next we establish the upper bound of g, where we need to prove \(\|\rho _{0}\theta \|_{L^{\infty }}\) and for the director field \(\| (\frac{\vert d_{y}\vert }{g} )^{2}\|_{L^{\infty }}\) and \(\|\frac{1}{\sqrt{\rho _{0}}} (\frac{\vert d_{y}\vert }{g} )^{2}\|_{\infty }^{2}\) type a priori estimate. The need of the weighted estimate for temperature and director field is necessary for the upper bound of g. As a preparation of deriving the a priori upper bound of g and the a priori \(L^{\infty }(0,T; L^{2})\cap L^{2}(0,T; H^{1})\) type estimates on \((u, d_{y}, \theta )\), for simplicity, the density-weighted estimates of θ and \(d_{y}\) are given in the following proposition.
Proposition 2
We have the following two items: (i) It holds that
$$\begin{aligned}& \biggl\Vert \frac{1}{\rho _{0}^{2}(y_{0})} \biggl(\frac{d_{y}}{g} \biggr)^{2} \biggr\Vert _{\infty }^{2}\leq C +C \biggl\Vert \frac{1}{\sqrt{g}}\partial _{y} \biggl(\frac{d_{y}}{g} \biggr)^{2} \biggr\Vert _{2}^{\frac{4}{3}} \Vert g \Vert _{\infty }^{\frac{2}{3}}, \\& \biggl\Vert \biggl(\frac{ \vert d_{y} \vert }{g} \biggr)^{2} \biggr\Vert _{\infty }\leq \biggl\Vert \frac{1}{\sqrt{g}}\partial _{y} \biggl( \frac{d_{y}}{g} \biggr)^{2} \biggr\Vert _{2}+C, \\& \bigl\Vert \rho _{0}^{2}\theta \bigr\Vert _{\infty }^{2} \leq C+C \biggl\Vert \frac{\partial _{y}\theta }{\sqrt{g}} \biggr\Vert _{2}^{\frac{4}{3}} \Vert g \Vert _{\infty }^{\frac{2}{3}}, \\& \Vert \theta \Vert _{\infty } \leq \sqrt{l} \biggl\Vert \frac{\partial _{y}\theta }{\sqrt{g}} \biggr\Vert _{2}+C, \end{aligned}$$
where C depends on \((\bar{\rho }_{0}, \rho _{0}^{\prime }, E_{0}, l,|\Omega _{0}| )\) and
$$ \bar{\rho }= \Vert \rho _{0} \Vert _{\infty },\qquad \Omega _{0}:= \biggl\{ y\in (0,l) \Big|\rho _{0}(y)\geq \frac{\bar{\rho }}{2} \biggr\} . $$
(ii) As a consequence of (i), we have
$$\begin{aligned}& \biggl\Vert \frac{1}{\sqrt{\rho _{0}}} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr\Vert _{\infty }^{2} \leq \eta \biggl\Vert \frac{1}{\sqrt{g}}\partial _{y} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr\Vert _{2}^{2}+C_{\eta } \bigl( \Vert g \Vert _{\infty }^{2}+1\bigr), \\& \Vert \sqrt{\rho _{0}}\theta \Vert _{\infty }^{2} \leq \eta \biggl\Vert \frac{\partial _{y}\theta }{\sqrt{g}} \biggr\Vert _{2}^{2}+C_{\eta } \bigl( \Vert g \Vert _{\infty }^{2}+1\bigr) \end{aligned}$$
for any \(\eta \in (0,\infty )\), where \(C_{\eta }\) is a positive constant depending only on η and \(\mathcal{N}_{1}(\bar{\rho _{0}},E_{0}, \Omega _{0}, \rho _{0}')\).
Proof
Now the proof of Proposition 2 is given as follows: (i) Let
$$ \biggl\vert \frac{d_{y}}{g}(y,t) \biggr\vert ^{2}= \mathcal{R}(y,t) $$
and
$$ \Omega _{0}:= \biggl\{ y\in (0,l)\Big|\rho _{0}(y)\geq \frac{\bar{\rho }}{2} \biggr\} . $$
Noticing that
$$ \mathcal{R}(y,t)=\frac{1}{\Omega _{0}} \int _{\Omega _{0}}\mathcal{R}\,dz+ \frac{1}{\Omega _{0}} \int _{\Omega _{0}} \int _{z}^{y}\partial _{y} \mathcal{R}\,dy \,dz, $$
we deduce that
$$\begin{aligned} \Vert \mathcal{R} \Vert _{\infty } \leq &\frac{1}{\Omega _{0}} \int _{\Omega _{0}} \mathcal{R}\,dz+ \int _{0}^{l} \vert \partial _{y} \mathcal{R} \vert \,dz \\ \leq &\frac{2}{\Omega _{0}} \Vert \mathcal{R} \Vert _{1}+ \biggl( \int _{0}^{l} \biggl\vert \frac{\partial _{y} \mathcal{R}}{\sqrt{g}} \biggr\vert ^{2}\,dz \biggr)^{ \frac{1}{2}} \biggl( \int _{0}^{l} g \,dz \biggr)^{\frac{1}{2}}, \end{aligned}$$
thus one can get
$$ \Vert \mathcal{R} \Vert _{\infty }\leq \sqrt{l} \biggl\Vert \frac{\partial _{y}\mathcal{R}}{\sqrt{g}} \biggr\Vert _{2}+\frac{2E_{0}}{\Omega _{0}}. $$
Density-weighted estimate for the director field:
By the Hölder and Young inequalities, and similarly as above, we deduce
$$\begin{aligned} \biggl(\frac{\mathcal{R}}{\rho _{0}^{2}} \biggr)^{2}(y,t) \leq & \Omega _{0}^{-1} \biggl(\frac{2}{\bar{\rho }^{2}} \biggr)^{2} \Vert \mathcal{R} \Vert _{1}^{2} +\frac{4}{\bar{\rho }^{3}} \biggl\Vert \frac{\mathcal{R}}{\rho _{0}^{2}} \biggr\Vert _{\infty } \Vert \mathcal{R} \Vert _{1} \bigl\Vert \rho _{0}' \bigr\Vert _{\infty } \\ &{}+\frac{2}{\bar{\rho }^{2}} \biggl\Vert \frac{\partial _{y}\mathcal{R}}{\sqrt{g}} \biggr\Vert _{2} \Vert g \Vert _{\infty }^{ \frac{1}{2}} \biggl\Vert \frac{\mathcal{R}}{\rho _{0}^{2}} \biggr\Vert _{2} \\ \leq &\Omega _{0}^{-1} \biggl(\frac{2}{\bar{\rho }^{2}} \biggr)^{2} \Vert \mathcal{R} \Vert _{1}^{2} + \frac{4}{\bar{\rho }^{3}} \biggl\Vert \frac{\mathcal{R}}{\rho _{0}^{2}(y_{0})} \biggr\Vert _{\infty } \Vert \mathcal{R} \Vert _{1} \bigl\Vert \rho _{0}' \bigr\Vert _{\infty } \\ &{}+\frac{2}{\bar{\rho }^{2}} \biggl\Vert \frac{\partial _{y}\mathcal{R}}{\sqrt{g}} \biggr\Vert _{2} \Vert g \Vert _{\infty }^{ \frac{1}{2}} \biggl\Vert \frac{\mathcal{R}}{\rho _{0}^{2}} \biggr\Vert _{1}^{\frac{1}{2}} \biggl\Vert \frac{\mathcal{R}}{\rho _{0}^{2}} \biggr\Vert _{\infty }^{\frac{1}{2}} \\ \leq &\Omega _{0}^{-1} \biggl(\frac{2}{\bar{\rho }^{2}} \biggr)^{2} \Vert \mathcal{R} \Vert _{1}^{2} + \frac{4}{\bar{\rho }^{3}} \biggl\Vert \frac{\mathcal{R}}{\rho _{0}^{2}(y_{0})} \biggr\Vert _{\infty } \Vert \mathcal{R} \Vert _{1} \bigl\Vert \rho _{0}' \bigr\Vert _{\infty } \\ &{}+\frac{2}{\bar{\rho }^{3}} \biggl\Vert \frac{\partial _{y}\mathcal{R}}{\sqrt{g}} \biggr\Vert _{2} \Vert g \Vert _{\infty }^{ \frac{1}{2}} \Vert \mathcal{R} \Vert _{1}^{\frac{1}{2}} \biggl\Vert \frac{\mathcal{R}}{\rho _{0}^{2}} \biggr\Vert _{\infty }^{\frac{1}{2}} \\ \leq &\delta \biggl\Vert \frac{\mathcal{R}}{\rho _{0}^{2}(y_{0})} \biggr\Vert _{\infty }^{2} +\Omega _{0}^{-1} \biggl( \frac{2}{\bar{\rho }^{2}} \biggr)^{2} \Vert \mathcal{R} \Vert _{1}^{2} +C(\delta ) \Vert \mathcal{R} \Vert _{1}^{2} \bigl\Vert \rho _{0}' \bigr\Vert _{\infty }^{2} \\ &{}+\frac{2}{\bar{\rho }^{4}} \biggl\Vert \frac{\partial _{y}\mathcal{R}}{\sqrt{g}} \biggr\Vert _{2}^{\frac{4}{3}} \Vert g \Vert _{\infty }^{\frac{2}{3}} \Vert \mathcal{R} \Vert _{1}^{\frac{2}{3}} \end{aligned}$$
for any \(y\in (0,l)\). Thus, from Lemma 1 and suitably small \(\delta >0\), we have
$$ \biggl\Vert \frac{\mathcal{R}}{\rho _{0}^{2}(y_{0})} \biggr\Vert _{\infty }^{2} \leq C +C \biggl\Vert \frac{\partial _{y}\mathcal{R}}{\sqrt{g}} \biggr\Vert _{2}^{ \frac{4}{3}} \Vert g \Vert _{\infty }^{\frac{2}{3}}. $$
(2.18)
(ii) Thanks to (i), we have
$$\begin{aligned} & \biggl\Vert \frac{\mathcal{R}}{\rho _{0}^{2}(y_{0})} \biggr\Vert _{\infty }^{2} \leq C +C \biggl\Vert \frac{\partial _{y}\mathcal{R}}{\sqrt{g}} \biggr\Vert _{2}^{ \frac{4}{3}} \Vert g \Vert _{\infty }^{\frac{2}{3}}, \\ & \Vert \mathcal{R} \Vert _{\infty }\leq \sqrt{l} \biggl\Vert \frac{\partial _{y}\mathcal{R}}{\sqrt{g}} \biggr\Vert _{2}+\frac{2E_{0}}{\Omega _{0}} \end{aligned}$$
for a positive constant C depending only on \(\mathcal{N}_{1}\). Therefore, we have
$$\begin{aligned} \biggl\Vert \frac{\mathcal{R}}{\sqrt{\rho _{0}}} \biggr\Vert _{\infty }^{2} =& \biggl\Vert \biggl( \frac{\mathcal{R}}{\rho _{0}^{2}} \biggr) ^{\frac{1}{4}} \mathcal{R}^{ \frac{3}{4}} \biggr\Vert _{\infty }^{2} \leq \biggl\Vert \biggl( \frac{\mathcal{R}}{\rho _{0}^{2}} \biggr) \biggr\Vert _{\infty }^{\frac{1}{2}} \Vert \mathcal{R} \Vert _{\infty }^{\frac{3}{2}} \\ \leq &C \biggl(1+ \biggl\Vert \frac{\partial _{y}\mathcal{R}}{\sqrt{g}} \biggr\Vert _{2}^{\frac{4}{3}} \Vert g \Vert _{\infty }^{\frac{2}{3}} \biggr)^{ \frac{1}{4}} \biggl( \biggl\Vert \frac{\partial _{y}\mathcal{R}}{\sqrt{g}} \biggr\Vert _{2}+1 \biggr)^{\frac{3}{2}}. \end{aligned}$$
Density-weighted estimate for temperature:
Assume \(\|\rho \|_{\infty }=\bar{\rho }\), let there exist \(y_{0}\in (0,l)\) such that
$$ \rho _{0}^{2}(y_{0})\theta (y_{0},t) \leq \frac{2}{l} \int _{0}^{l}\rho _{0}^{2} \theta \,d\xi \leq \frac{2\bar{\rho }}{l} \Vert \rho _{0}\theta \Vert _{1}. $$
By the Hölder and Young inequalities, we deduce
$$\begin{aligned} \bigl(\rho _{0}^{2}\theta \bigr)^{2}(y,t)\leq{}& ( \rho _{0}\theta )^{2}(y_{0},t)+2 \int _{0}^{l}\rho _{0}^{2} \theta \bigl\vert \partial _{y}\bigl(\rho _{0}^{2} \theta \bigr) \bigr\vert d \xi \\ \leq{}& \biggl(\frac{2\bar{\rho }}{l} \biggr)^{2} \Vert \rho _{0}\theta \Vert _{1}^{2}+2 \int _{0}^{l}\bigl(2\rho _{0}^{2} \theta \rho _{0}\theta \bigl\vert \rho _{0}' \bigr\vert +\rho _{0}^{2} \theta \rho _{0}^{2} \vert \partial _{y}\theta \vert \bigr)\,d\xi \\ \leq {}&\biggl(\frac{2\bar{\rho }}{l} \biggr)^{2} \Vert \rho _{0}\theta \Vert _{1}^{2}+4 \bigl\Vert \rho _{0}^{2}\theta \bigr\Vert _{\infty } \Vert \rho _{0}\theta \Vert _{1} \bigl\Vert \rho _{0}' \bigr\Vert _{\infty } \\ & {}+2\bar{\rho }^{2} \biggl\Vert \frac{\partial _{y}\theta }{\sqrt{g}} \biggr\Vert _{2} \Vert g \Vert _{\infty }^{\frac{1}{2}} \bigl\Vert \rho _{0}^{2}\theta \bigr\Vert _{2} \\ \leq{}& \biggl(\frac{2\bar{\rho }}{l} \biggr)^{2} \Vert \rho _{0}\theta \Vert _{1}^{2}+4 \bigl\Vert \rho _{0}^{2}\theta \bigr\Vert _{\infty } \Vert \rho _{0}\theta \Vert _{1} \bigl\Vert \rho _{0}' \bigr\Vert _{\infty }+ +4 \bigl\Vert \rho _{0}^{2}\theta \bigr\Vert _{\infty } \Vert \rho _{0}\theta \Vert _{1} \bigl\Vert \rho _{0}' \bigr\Vert _{\infty } \\ &{}+2\bar{\rho }^{2} \biggl\Vert \frac{\partial _{y}\theta }{\sqrt{g}} \biggr\Vert _{2} \Vert g \Vert _{\infty }^{\frac{1}{2}} \bigl\Vert \rho _{0}^{2}\theta \bigr\Vert _{1}^{\frac{1}{2}} \bigl\Vert \rho _{0}^{2} \theta \bigr\Vert _{\infty }^{\frac{1}{2}}+2\bar{\rho }^{\frac{5}{2}} \biggl\Vert \frac{\partial _{y}\theta }{\sqrt{g}} \biggr\Vert _{2} \Vert g \Vert _{\infty }^{ \frac{1}{2}} \Vert \rho _{0}\theta \Vert _{1}^{\frac{1}{2}} \bigl\Vert \rho _{0}^{2} \theta \bigr\Vert _{\infty }^{\frac{1}{2}} \\ \leq{}& \frac{1}{2} \bigl\Vert \rho _{0}^{2}\theta \bigr\Vert _{\infty }^{2} + \biggl( \frac{2\bar{\rho }}{l} \biggr)^{2} \Vert \rho _{0}\theta \Vert _{1}^{2} +16 \Vert \rho _{0}\theta \Vert _{1}^{2} \bigl\Vert \rho _{0}' \bigr\Vert _{\infty }^{2} \\ &{}+3\bar{\rho }^{\frac{10}{3}} \Vert \rho _{0}\theta \Vert _{1}^{\frac{2}{3}} \biggl\Vert \frac{\partial _{y}\theta }{\sqrt{g}} \biggr\Vert _{2}^{\frac{4}{3}} \Vert g \Vert _{\infty }^{\frac{2}{3}} \end{aligned}$$
for any \(y\in (0,l)\), it is not hard to see that
$$ \bigl\Vert \rho _{0}^{2}\theta \bigr\Vert _{\infty }^{2}\leq C + C \biggl\Vert \frac{\partial _{y}\theta }{\sqrt{g}} \biggr\Vert _{2}^{\frac{4}{3}} \Vert g \Vert _{\infty }^{\frac{2}{3}}. $$
Noticing that
$$ \theta (y,t)=\frac{1}{\Omega _{0}} \int _{\Omega _{0}}\theta \,dz+ \frac{1}{\Omega _{0}} \int _{\Omega _{0}} \int _{z}^{y}\partial _{y} \theta \,d\xi \,dz, $$
we deduce, by the Hölder inequality, that
$$\begin{aligned} \Vert \theta \Vert _{\infty } \leq &\frac{1}{\Omega _{0}} \int _{\Omega _{0}} \frac{\rho _{0}\theta }{\rho _{0}}\,dz+ \int _{0}^{l} \vert \partial _{y} \theta \vert \,dz \\ \leq &\frac{2}{\Omega _{0}\bar{\rho }} \Vert \rho _{0}\theta \Vert _{1}+ \biggl( \int _{0}^{l} \biggl\vert \frac{\partial _{y}\theta }{\sqrt{g}} \biggr\vert ^{2}\,dz \biggr)^{\frac{1}{2}} \biggl( \int _{0}^{l} g \,dz \biggr)^{\frac{1}{2}}. \end{aligned}$$
□
The other estimates for θ in Proposition 2 can be obtained similar to the director field.
By density-weighted estimate of θ and \(d_{y}\) in hand, the desired a priori \(L^{\infty }(0,T; L^{2})\cap L^{2}(0,T; H^{1})\) estimates on \((u, \frac{d_{y}}{g}, \theta )\) are given as in the following lemma.
Lemma 2
Given \(T\in (0,\infty )\). It holds that
$$\begin{aligned} &\frac{d}{dt} \biggl( \Vert \sqrt{\rho _{0}} \mathcal{E} \Vert _{2}^{2} + \frac{C_{1}}{\mu } \bigl\Vert \sqrt{\rho _{0}}u^{2} \bigr\Vert _{2}^{2} + \biggl\Vert \bigg\vert \frac{d_{y}}{g^{\frac{3}{4}}}\bigg\vert ^{2} \biggr\Vert _{2}^{2} \biggr) \\ &\qquad {} + \int _{0}^{t} \biggl( \Vert \sqrt{\rho _{0}}\theta \Vert _{\infty }^{2} + \frac{1}{\sqrt{\rho _{0}}} \biggl\vert \frac{ d_{y}}{g} \biggr\vert ^{2} \| _{\infty }^{2} \biggr)\,d\tau \\ &\qquad {} + \int _{0}^{l} \biggl[ \biggl( \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr)_{y} \biggr]^{2}\,dy+5C_{1} \biggl\Vert \frac{u\partial _{y}u}{\sqrt{g}} \biggr\Vert _{2}^{2}+\kappa c_{v} \biggl\Vert \frac{\partial _{y}\theta }{\sqrt{g}} \biggr\Vert _{2}^{2} \\ &\quad \leq \frac{C}{\eta } \biggl[(e^{\int _{0}^{t} \Vert \frac{ \vert d_{y} \vert }{g} \Vert _{L^{\infty }}^{2}\,d\tau } \biggl(1+ \frac{R}{\mu } \int _{0}^{t} \Vert \rho _{0}\theta \Vert _{\infty }\,d\tau \biggr) \biggr]^{2}+\frac{C}{2} \frac{d}{dt} \int _{0}^{l}\rho _{0}\bigl( \mathcal{E}^{ \prime }\bigr)^{2}\,dy \end{aligned}$$
(2.19)
for any \(t\in (0,T)\).
Denote \(\mathcal{E}:=\frac{1}{2}(u^{2}+ \frac{1}{\rho _{0}} \frac{\vert d_{y}\vert ^{2}}{g})+c_{v}\theta \), then one can derive from (1.8)2, (1.8)3, and (1.8)4 that
$$\begin{aligned}& \rho _{0}\partial _{t}\mathcal{E}+\partial _{y} \biggl(u\biggl(p+ \frac{1}{2g^{2}} \vert d_{y} \vert ^{2}\biggr) \biggr)-\kappa \partial _{y} \biggl( \frac{\partial _{y}\theta }{g} \biggr) \\& \quad =\mu \partial _{y} \biggl(\frac{1}{g} \partial _{y} \biggl(\frac{u^{2}}{2} \biggr) \biggr) + \biggl(d_{t} \cdot \frac{d_{y}}{g} \biggr)_{y}. \end{aligned}$$
(2.20)
Multiplying (2.20) by \(\mathcal{E}^{\prime }=\frac{1}{2}(u^{2}+ \frac{\vert d_{y}\vert ^{2}}{g^{2}})+c_{v}\theta \) and integrating the resultant over \((0,l)\), one gets from integration by parts that
$$\begin{aligned} \begin{aligned}&\frac{d}{dt} \int _{0}^{l}\bigl(\rho _{0}\mathcal{E} \mathcal{E}^{\prime }\bigr)\,dy+ \int _{0}^{l}\frac{1}{g} \biggl( \kappa \partial _{y}\theta +\mu u \partial _{y} u + \biggl(d_{t}\cdot \frac{d_{y}}{g} \biggr) \biggr) \partial _{y}\mathcal{E}^{\prime }\,dy \\ &\quad = \int _{0}^{l} \rho _{0}\mathcal{E} \bigl(\mathcal{E}^{\prime }\bigr)_{t}\,dy + \int _{0}^{l} u \biggl(p+\frac{1}{2g^{2}} \vert d_{y} \vert ^{2} \biggr)\partial _{y} \mathcal{E}^{\prime }\,dy, \\ &\frac{d}{dt} \int _{0}^{l}\bigl(\rho _{0}\mathcal{E} \mathcal{E}^{\prime }\bigr)\,dy+ \int _{0}^{l}\frac{1}{g} \biggl( \kappa \partial _{y}\theta +\mu u \partial _{y} u + \biggl( \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr)_{y} \biggr)\partial _{y}\mathcal{E}^{\prime }\,dy \\ &\quad \leq \frac{C}{2}\frac{d}{dt} \int _{0}^{l}\rho _{0}\bigl( \mathcal{E}^{\prime }\bigr)^{2}\,dy + \int _{0}^{l} u \biggl(p+ \frac{1}{2g^{2}} \vert d_{y} \vert ^{2} \biggr)\partial _{y} \mathcal{E}^{\prime }\,dy, \end{aligned} \end{aligned}$$
(2.21)
where we have used that \(\mathcal{E} \leq \mathcal{E}^{\prime } \) and the fact \(d_{t}\cdot \frac{d_{y}}{g}= \frac{1}{g} ( \vert \frac{d_{y}}{g} \vert ^{2} )_{y}\). By the Young inequality, we have
$$\begin{aligned} &\int _{0}^{l}\frac{1}{g} \biggl( \kappa \partial _{y}\theta +\mu u \partial _{y} u+ \biggl( \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr)_{y} \biggr)\partial _{y}\mathcal{E}^{\prime }\,dy \\ &\quad = \int _{0}^{l}\frac{1}{g} \biggl(\kappa \partial _{y}\theta +\mu u \partial _{y} u+ \biggl( \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr)_{y} \biggr) \biggl[u\partial _{y} u+ \biggl( \frac{ \vert d_{y} \vert ^{2}}{g^{2}} \biggr)_{y}+ c_{v}\partial _{y} \theta \biggr]\,dy \\ &\quad \geq \frac{3\kappa c_{v}}{4} \int _{0}^{l} \biggl\vert \frac{\partial _{y}\theta }{\sqrt{g}} \biggr\vert ^{2}\,dy-C \int _{0}^{l} \biggl[ \biggl\vert \frac{u\partial _{y}u}{\sqrt{g}} \biggr\vert ^{2} + \biggl\vert \biggl( \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr)_{y} \biggr\vert ^{2} \biggr]\,dy \end{aligned}$$
and
$$\begin{aligned} &\int _{0}^{l} u \biggl(p+\frac{1}{2g^{2}} \vert d_{y} \vert ^{2} \biggr)\partial _{y} \mathcal{E}\,dy \\ &\quad =R \int _{0}^{l} u\frac{\rho _{0}}{g}\theta \biggl(u \partial _{y}u + \biggl( \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr)_{y} + c_{v}\partial _{y} \theta \biggr)\,dy \\ &\qquad {} + \int _{0}^{l} u \frac{1}{2g^{2}} \vert d_{y} \vert ^{2} \biggl(u \partial _{y}u + \biggl( \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr)_{y} + c_{v}\partial _{y}\theta \biggr)\,dy \\ &\quad \leq \frac{\kappa c_{v}}{4} \int _{0}^{l} \biggl\vert \frac{\partial _{y}\theta }{\sqrt{g}} \biggr\vert ^{2}\,dy +C \int _{0}^{l}\frac{1}{g} \biggl(\rho _{0}^{2}u^{2} \theta ^{2}+(u \partial _{y}u)^{2} + \biggl[ \biggl( \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr)_{y} \biggr]^{2} \biggr)\,dy, \\ &\qquad {}+\frac{\kappa c_{v}}{4} \int _{0}^{l} \biggl\vert \frac{\partial _{y}\theta }{\sqrt{g}} \biggr\vert ^{2}\,dy +C \int _{0}^{l}\frac{1}{g} \biggl(u^{2} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{4}+(u \partial _{y}u)^{2} + \biggl[ \biggl( \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr)_{y} \biggr]^{2} \biggr)\,dy, \end{aligned}$$
for a positive constant C depending only on R, \(c_{v}\), μ, and κ. Substituting the above two inequalities into (2.21) and applying Lemma 1 and Proposition 1, we obtain
$$\begin{aligned} &\frac{d}{dt} \Vert \sqrt{\rho _{0}}\mathcal{E} \Vert _{2}^{2}+\kappa c_{v} \biggl\Vert \frac{\partial _{y}\theta }{\sqrt{g}} \biggr\Vert _{2}^{2} \\ &\quad \leq \delta \biggl( \biggl\Vert \frac{u\partial _{y}u}{\sqrt{g}} \biggr\Vert _{2}^{2}+ \biggl\Vert \frac{1}{\sqrt{g}} \biggl( \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr)_{y} \biggr\Vert _{2}^{2} \biggr) \\ &\qquad {}+ C \biggl( \int _{0}^{l}\frac{\rho _{0}u^{2}}{g}\rho _{0} \theta ^{2} + \frac{1}{g} \biggl(u^{2} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{4} \biggr)\,dy \biggr) + \frac{C}{2}\frac{d}{dt} \int _{0}^{l}\rho _{0}\bigl( \mathcal{E}^{\prime }\bigr)^{2}\,dy \\ &\quad \leq \delta \biggl( \biggl\Vert \frac{u\partial _{y}u}{\sqrt{g}} \biggr\Vert _{2}^{2}+ \biggl\Vert \frac{1}{\sqrt{g}} \biggl( \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr)_{y} \biggr\Vert _{2}^{2} \biggr) \\ &\qquad {}+C \Vert \sqrt{\rho _{0}}u \Vert _{2}^{2} \biggl( \Vert \sqrt{\rho _{0}}\theta \Vert _{\infty }^{2} + \biggl\Vert \frac{1}{\sqrt{\rho _{0}}} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr\Vert _{\infty }^{2} \biggr)+ \frac{C}{2}\frac{d}{dt} \int _{0}^{l}\rho _{0}\bigl( \mathcal{E}^{\prime }\bigr)^{2}\,dy \\ &\quad \leq \delta \biggl( \biggl\Vert \frac{u\partial _{y}u}{\sqrt{g}} \biggr\Vert _{2}^{2}+ \biggl\Vert \frac{1}{\sqrt{g}} \biggl( \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr)_{y} \biggr\Vert _{2}^{2} \biggr) \\ &\qquad {}+C \biggl( \Vert \sqrt{\rho _{0}}\theta \Vert _{\infty }^{2} + \biggl\Vert \frac{1}{\sqrt{\rho _{0}}} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr\Vert _{\infty }^{2} \biggr) + \frac{C}{2}\frac{d}{dt} \int _{0}^{l}\rho _{0}\bigl( \mathcal{E}^{ \prime }\bigr)^{2}\,dy \end{aligned}$$
for a positive constant C depending only on R, \(c_{v}\), μ, and κ, and thus
$$\begin{aligned}& \frac{d}{dt} \Vert \sqrt{\rho _{0}}\mathcal{E} \Vert _{2}^{2}+\kappa c_{v} \biggl\Vert \frac{\partial _{y}\theta }{\sqrt{g}} \biggr\Vert _{2}^{2} \end{aligned}$$
(2.22)
$$\begin{aligned}& \quad \leq C_{1} \biggl( \biggl\Vert \frac{u\partial _{y} u}{\sqrt{g}} \biggr\Vert _{2}^{2} + \biggl\Vert \frac{1}{\sqrt{g}} \biggl( \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr)_{y} \biggr\Vert _{2}^{2} \biggr) +C \biggl( \Vert \sqrt{ \rho _{0}}\theta \Vert _{\infty }^{2} + \biggl\Vert \frac{1}{\sqrt{\rho _{0}}} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr\Vert _{\infty }^{2} \biggr) \end{aligned}$$
(2.23)
$$\begin{aligned}& \qquad {} + \frac{C}{2}\frac{d}{dt} \int _{0}^{l}\rho _{0}\bigl( \mathcal{E}^{\prime }\bigr)^{2}\,dy \end{aligned}$$
(2.24)
for a positive constant \(C_{1}\) depending only on R, \(c_{v}\), μ, and κ.
Multiplying the momentum equation by \(4u^{3}\) and integrating the resultant over \((0,l)\), after integration by parts and the Young inequality, one gets that
$$\begin{aligned} &\frac{d}{dt} \int _{0}^{l}\rho _{0} u^{4}\,dy+12\mu \int _{0}^{l} \biggl\vert \frac{u\partial _{y}u}{\sqrt{g}} \biggr\vert ^{2}\,dy \\ &\quad =12 \int _{0}^{l} \biggl(p+\frac{1}{2g^{2}} \vert d_{y} \vert ^{2} \biggr) u^{2} \partial _{y}u\,dy \\ &\quad =12 \int _{0}^{l} \biggl(\frac{R\rho _{0}}{g} \theta + \frac{1}{2g^{2}} \vert d_{y} \vert ^{2} \biggr) u^{2}\partial _{y}u\,dy \\ &\quad \leq 6\mu \int _{0}^{l} \biggl\vert \frac{u\partial _{y}u}{\sqrt{g}} \biggr\vert ^{2}\,dy + \frac{6R^{2}}{\mu } \int _{0}^{l}\frac{\rho _{0}u^{2}}{g}\rho _{0} \theta ^{2}\,dy + \frac{6}{\mu } \int _{0}^{l}\frac{1}{g} \biggl(u^{2} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{4} \biggr)\,dy, \\ &\quad \leq C_{1} \biggl( \biggl\Vert \frac{u\partial _{y}u}{\sqrt{g}} \biggr\Vert _{2}^{2} \biggr)+ C \Vert \sqrt{\rho _{0}}u \Vert _{2}^{2} \biggl( \Vert \sqrt{\rho _{0}} \theta \Vert _{\infty }^{2} + \biggl\Vert \frac{1}{\sqrt{\rho _{0}}} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr\Vert _{\infty }^{2} \biggr) \\ &\quad \leq C_{1} \biggl( \biggl\Vert \frac{u\partial _{y}u}{\sqrt{g}} \biggr\Vert _{2}^{2} \biggr) + C \biggl( \Vert \sqrt{\rho _{0}}\theta \Vert _{\infty }^{2} + \biggl\Vert \frac{1}{\sqrt{\rho _{0}}} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr\Vert _{\infty }^{2} \biggr), \end{aligned}$$
from which, by Proposition 1, the Hölder inequality, and Proposition 1, one obtains
$$\begin{aligned} \frac{d}{dt} \int _{0}^{l}\rho _{0} u^{4}\,dy+6\mu \int _{0}^{l} \biggl\vert \frac{u\partial _{y}u}{\sqrt{g}} \biggr\vert ^{2}\,dy \leq & C \biggl( \Vert \sqrt{ \rho _{0}}\theta \Vert _{\infty }^{2} + \biggl\Vert \frac{1}{\sqrt{\rho _{0}}} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr\Vert _{\infty }^{2} \biggr), \end{aligned}$$
that is,
$$\begin{aligned} \frac{d}{dt} \bigl\Vert \sqrt{\rho _{0}} u^{2} \bigr\Vert _{2}^{2} + 6\mu \biggl\Vert \frac{u\partial _{y} u}{\sqrt{g}} \biggr\Vert _{2}^{2} \leq C \biggl( \Vert \sqrt{\rho _{0}}\theta \Vert _{\infty }^{2} + \biggl\Vert \frac{1}{\sqrt{\rho _{0}}} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr\Vert _{\infty }^{2} \biggr). \end{aligned}$$
(2.25)
Now we first operate the \(\partial _{y}\) on both sides of the director field equation
$$\begin{aligned} (d_{y})_{t}= \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y}+ \biggl( \frac{ \vert {d}_{y} \vert ^{2}}{g^{2}}d\biggr)_{y}. \end{aligned}$$
Taking the dot product of the equation by \((\frac{\vert d_{y}\vert ^{2}}{g^{3}}d_{y} )\), we get
$$\begin{aligned}& (d_{y})_{t}\cdot \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2}\frac{d_{y}}{g}= \biggl( \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr)_{y}\cdot \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2}\frac{d_{y}}{g}+ \biggl( \frac{ \vert {d}_{y} \vert ^{2}d}{g^{2}} \biggr)_{y}\cdot \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2}\frac{d_{y}}{g}. \end{aligned}$$
(2.26)
Now we compute each term on the left- and right-hand side of equation (2.26) one by one. The term on the left-hand side after integration by parts can be written as
$$\begin{aligned} \int _{0}^{l}(d_{y})_{t}\cdot \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \frac{d_{y}}{g}\,dy&= \frac{1}{4} \int _{0}^{l} \biggl(d_{y}\cdot \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2}\frac{d_{y}}{g} \biggr)_{t}\,dy+\frac{3}{4} \int _{0}^{l} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{4}g_{t}\,dy \\ &=\frac{1}{4} \int _{0}^{l} \biggl(\frac{ \vert d_{y} \vert ^{4}}{g^{3}} \biggr)_{t}\,dy+\frac{3}{4} \int _{0}^{l} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{4}u_{y}\,dy \\ &=\frac{1}{4} \int _{0}^{l} \biggl(\frac{ \vert d_{y} \vert ^{4}}{g^{3}} \biggr)_{t}\,dy - \frac{3}{4} \int _{0}^{l} \biggl( \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{4} \biggr)_{y}u \,dy \\ &=\frac{1}{4} \int _{0}^{l} \biggl(\frac{ \vert d_{y} \vert ^{4}}{g^{3}} \biggr)_{t}\,dy - \frac{3}{2} \int _{0}^{l} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggl( \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr)_{y}u \,dy \\ &=\frac{1}{4} \int _{0}^{l} \biggl(\frac{ \vert d_{y} \vert ^{4}}{g^{3}} \biggr)_{t}\,dy - \frac{3}{2} \int _{0}^{l}\frac{1}{\sqrt{\rho _{0}}} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggl( \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr)_{y}\sqrt{\rho _{0}}u \,dy. \end{aligned}$$
In the above calculation we have used the boundary conditions \(d_{y}=0\), \(u=0\). For the terms on the right-hand side, we calculate them one by one. The first term after integration by parts and using boundary condition is as follows:
$$\begin{aligned} &\int _{0}^{l} \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y}\cdot \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2}\frac{d_{y}}{g}\,dy \\ &\quad = - \int _{0}^{l}\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y}\cdot \biggl( \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2}\frac{d_{y}}{g} \biggr)_{y}\,dy \\ &\quad =- \int _{0}^{l}\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y}\cdot \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggl(\frac{d_{y}}{g} \biggr)_{y}\,dy - \int _{0}^{l}\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y}\cdot \biggl( \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr)_{y} \frac{d_{y}}{g}\,dy \\ &\quad =- \int _{0}^{l}\frac{1}{g} \biggl\vert \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\vert ^{2} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2}\,dy -\frac{1}{2} \int _{0}^{l} \biggl[ \biggl( \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr)_{y} \biggr]^{2}\,dy \\ &\quad \leq - \int _{0}^{l}\frac{1}{g} \biggl[ \biggl( \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr)_{y} \biggr]^{2}\,dy. \end{aligned}$$
Now the last term of (2.26) can be written as
$$\begin{aligned} \biggl(\frac{ \vert {d}_{y} \vert ^{2}d}{g^{2}} \biggr)_{y}\cdot \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2}\frac{d_{y}}{g}= \frac{ \vert {d}_{y} \vert ^{2}}{g^{2}}\cdot \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \frac{ \vert d_{y} \vert ^{2}}{g}, \end{aligned}$$
where we have used the constraint \(\vert d\vert ^{2}=1\) for the director field. Substituting the computed term in (2.26), integrating the resultant with respect to space variable, and using Holder’s inequality result in
$$\begin{aligned}& \frac{d}{dt} \biggl\Vert \biggl(\frac{d_{y}}{g^{\frac{3}{4}}} \biggr)^{2} \biggr\Vert _{2}^{2} + \biggl\Vert \frac{1}{\sqrt{g}} \biggl( \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr)_{y} \biggr\Vert _{2}^{2} \\& \quad \leq C \biggl( \biggl\Vert \frac{d_{y}}{\sqrt{g}} \biggr\Vert _{2}^{2} + \Vert \sqrt{ \rho _{0}}u \Vert _{2}^{2} \biggr) \biggl\Vert \frac{1}{\sqrt{\rho _{0}}} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr\Vert _{\infty }^{2}. \end{aligned}$$
(2.27)
Adding (2.27) and (2.25) with (2.24), one can get
$$\begin{aligned} &\frac{d}{dt} \biggl( \Vert \sqrt{\rho _{0}} \mathcal{E} \Vert _{2}^{2} + \frac{C_{1}}{\mu } \bigl\Vert \sqrt{\rho _{0}}u^{2} \bigr\Vert _{2}^{2} + \biggl\Vert \biggl( \frac{d_{y}}{g^{\frac{3}{4}}} \biggr)^{2} \biggr\Vert _{2}^{2} \biggr) \\ &\qquad {} + \biggl\Vert \frac{1}{\sqrt{g}} \biggl( \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr)_{y} \biggr\Vert _{2}^{2}+5C_{1} \biggl\Vert \frac{u\partial _{y}u}{\sqrt{g}} \biggr\Vert _{2}^{2}+\kappa c_{v} \biggl\Vert \frac{\partial _{y}\theta }{\sqrt{g}} \biggr\Vert _{2}^{2} \\ &\quad \leq C \biggl( \Vert \sqrt{\rho _{0}}\theta \Vert _{\infty }^{2} + \biggl\Vert \frac{1}{\sqrt{\rho _{0}}} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr\Vert _{\infty }^{2} \biggr)+ \frac{C}{2}\frac{d}{dt} \int _{0}^{l}\rho _{0}\bigl( \mathcal{E}^{ \prime }\bigr)^{2}\,dy \end{aligned}$$
(2.28)
for any \(t\in (0,T)\).
By Proposition 1 and (ii) of Proposition 2, we have
$$\begin{aligned} &\Vert \sqrt{\rho _{0}}\theta \Vert _{\infty }^{2}(t)+ \biggl\Vert \frac{1}{\sqrt{\rho _{0}}} \biggl\vert \frac{ d_{y}}{g} \biggr\vert ^{2} \biggr\Vert _{\infty }^{2}(t) \\ &\quad \leq \eta \biggl\Vert \frac{\partial _{y}\theta }{\sqrt{g}} \biggr\Vert _{2}^{2}(t) + \eta \biggl\Vert \frac{1}{\sqrt{g}} \biggl( \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr)_{y} \biggr\Vert _{2}^{2} \\ &\qquad {} +\frac{C}{\eta } \biggl(e^{\int _{0}^{t} \Vert ( \frac{ \vert d_{y} \vert }{g} )^{2} \Vert _{L^{\infty }}\,d\tau } \biggl(1+ \frac{R}{\mu } \int _{0}^{t} \Vert \rho _{0}\theta \Vert _{\infty }\,d\tau \biggr) \biggr)^{2}. \end{aligned}$$
(2.29)
Collecting the estimates provides the proof of the lemma.
In order to close the estimates of the lemma, we need to control the unbounded operator \(e^{\int _{0}^{t}\| (\frac{\vert d_{y}\vert }{g} )^{2}\|_{L^{ \infty }}\,d\tau }\) and \(\frac{R}{\mu }\int _{0}^{t}\|\rho _{0}\theta \|_{\infty }\,d\tau \). We see that the director field term is in product with the time integral of \(L^{\infty }-\mathit{norm} \) of temperature, so it cannot be controlled by right-hand side terms, thus we need new estimates to control these terms. In order to control the right-hand side of the above inequality, dissipation estimates are obtained on the director field in terms of \(L^{\infty }-\mathit{norm} \) of temperature.
Lemma 3
Given \(T\in (0,\infty )\). It holds that
$$\begin{aligned}& \sup_{0\leq t\leq T} \biggl\Vert \biggl(\sqrt{\rho _{0}}u, \frac{ \vert d_{y} \vert }{\sqrt{g}},\sqrt{\rho _{0}}\theta \biggr) \biggr\Vert _{2}^{2} + \int _{0}^{T} \biggl( \Vert \theta \Vert _{\infty }^{2}+ \biggl\Vert \biggl(\partial _{y} \theta ,\partial _{y} u, \frac{1}{\sqrt{g}} \biggl(\frac{d_{y}}{g} \biggr)_{y}\biggr) \biggr\Vert _{2}^{2} \biggr)\,dt \\& \quad \leq C \bigl((1+ T)+ \bigl\Vert \bigl(\sqrt{\rho _{0}}u_{0}, \sqrt{\rho _{0}}u_{0}^{2}, \sqrt{\rho _{0}} \theta _{0} \bigr) \bigr\Vert _{2}^{2} \bigr) + \int _{0}^{T} \Vert \theta \Vert _{\infty }\,d\tau \end{aligned}$$
for positive constants C depending only on R, \(c_{v}\), μ, κ, \(m_{1}\), \(\mathcal{N}_{1}\), and T.
Proof
Multiplying (1.8)2 by u and taking the dot product of director field equation (1.8)3 with \(- (\frac{1}{g} (\frac{d_{y}}{g} )_{y} + \frac{\vert {d}_{y}\vert ^{2}d}{g^{2}} )\) respectively, the resultant is integrated over \((0,l)\). From the momentum equation it is not hard to see that
$$\begin{aligned}& \frac{1}{2}\frac{d}{dt} \Vert \sqrt{ \rho _{0}}u \Vert _{2}^{2}+\mu \biggl\Vert \frac{\partial _{y} u}{\sqrt{g}} \biggr\Vert _{2}^{2} \\& \quad = \int _{0}^{l} \biggl(p + \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr)\partial _{y} u \,dy \\& \quad = R \int _{0}^{l} \biggl(\frac{\rho _{0}}{g}\theta + \biggl\vert \frac{d_{y}}{2g} \biggr\vert ^{2} \biggr)\partial _{y}u\,dy \\& \quad \leq \frac{\mu }{2} \biggl\Vert \frac{\partial _{y} u}{\sqrt{g}} \biggr\Vert _{2}^{2}+ \frac{R}{2\mu } \int _{0}^{l}\frac{\rho _{0}^{2}}{g}\theta ^{2}\,dy \\& \qquad {} + \frac{1}{\mu } \int _{0}^{l} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{4}\,dy. \end{aligned}$$
(2.30)
Similarly, from the director field equation, we have
$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \int _{0}^{1}\frac{ \vert d_{y} \vert ^{2}}{g}\,dy+ \int _{0}^{l} g \biggl\vert \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y}+ \frac{1}{g^{2}} \vert d_{y} \vert ^{2}d\biggr\vert ^{2}\,dy=- \int _{0}^{l} \frac{1}{2g^{2}} \vert d_{y} \vert ^{2} u_{y}\,dy. \end{aligned}$$
(2.31)
By using the constraint \(\vert d\vert ^{2}=1\), it is not not hard to see that
$$ g \biggl\vert \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y}+ \frac{1}{g^{2}} \vert d_{y} \vert ^{2}d\biggr\vert ^{2}=\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y}^{2}- \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{4}. $$
Now equation (2.31), can be written as
$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \int _{0}^{l}\frac{ \vert d_{y} \vert ^{2}}{g}\,dy+ \int _{0}^{l} \frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y}^{2}\,dy&= - \int _{0}^{l} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} u_{y}\,dy + \int \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{4} \\ &\leq \frac{\mu }{2} \biggl\Vert \frac{\partial _{y} u}{\sqrt{g}} \biggr\Vert _{2}^{2} + C \int \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{4}. \end{aligned}$$
(2.32)
Combining (2.30) and (2.32) and using Sobolev embedding result in
$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \biggl( \Vert \sqrt{\rho _{0}}u \Vert _{2}^{2}+ \biggl\Vert \frac{ \vert d_{y} \vert }{\sqrt{g}} \biggr\Vert _{2}^{2} \biggr)+\mu \biggl\Vert \frac{\partial _{y} u}{\sqrt{g}} \biggr\Vert _{2}^{2} + \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{2}^{2}, \\ &\quad \leq \frac{\mu }{2} \biggl\Vert \frac{\partial _{y} u}{\sqrt{g}} \biggr\Vert _{2}^{2} +\frac{R}{2\mu } \int _{0}^{l}\frac{\rho _{0}^{2}}{g}\theta ^{2}\,dy+ \frac{1}{2} \biggl\Vert \frac{1}{\sqrt{g}} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{2}^{2} + C. \end{aligned}$$
(2.33)
The constraint \(\vert d\vert ^{2}=1\) is used in the above inequality, thus, by Proposition 1,
$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \biggl( \Vert \sqrt{\rho _{0}}u \Vert _{2}^{2}+ \biggl\Vert \frac{ \vert d_{y} \vert }{\sqrt{g}} \biggr\Vert _{2}^{2} \biggr)+\mu \biggl\Vert \frac{\partial _{y} u}{\sqrt{g}} \biggr\Vert _{2}^{2} + \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{2}^{2} \\ &\quad \leq C\bigl( \Vert \rho _{0} \Vert _{\infty } \bigr) \bigl( \Vert \theta \Vert _{\infty }+ 1 \bigr). \end{aligned}$$
(2.34)
□
Thus we see that, in order to close the above estimates, we need a control on temperature, which will be given in the following lemma.
: Now we prove the following key lemma.
Lemma 4
For any \(t\geq 0\), it holds that
$$ \int _{0}^{t} \Vert \theta \Vert _{L^{\infty }}\,d\tau \leq C(1+ t)^{2}, $$
whereas C depends on \(\|\rho _{0}\|_{\infty }\) and the initial data.
Proof
Multiplying the temperature equation by \((\theta + \delta )^{-1}\) for some \(\delta \in (0,1)\), the resultant equation can be written as
$$\begin{aligned} \rho _{0} \bigl(\ln (\theta + \delta ) \bigr)_{t} + \frac{\rho _{0}\theta }{(\theta +\delta )g}u_{y}={}& \kappa \biggl( \frac{\theta _{y}}{(\theta + \delta )g} \biggr)_{y}+\kappa \biggl( \frac{\theta _{y}^{2}}{(\theta + \delta )^{2}g} \biggr) \\ &{}+ \biggl(\frac{u_{y}^{2}}{g}+ g \biggl\vert \frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} + \frac{1}{g^{2}} \vert d_{y} \vert ^{2}d\biggr\vert ^{2} \biggr) \frac{1}{\theta + \delta }. \end{aligned}$$
(2.35)
Integrating the above inequality over time and space variables for \((\theta + \delta ) \geq 1\), then using boundary condition for temperature, we get
$$\begin{aligned} & \int _{0}^{l}\rho _{0} \bigl(\ln (\theta + \delta ) \bigr) + \int _{0}^{l} \int _{0}^{t}\kappa \biggl( \frac{\theta _{y}^{2}}{(\theta + \delta )^{2}g} \biggr)\,dy\,d\tau \\ &\qquad {} + \int _{0}^{l} \int _{0}^{t} \biggl(\frac{u_{y}^{2}}{g}+ g \biggl\vert \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} + \frac{1}{g^{2}} \vert d_{y} \vert ^{2}d\biggr\vert ^{2} \biggr) \frac{1}{\theta + \delta }\,dy\,d\tau \\ &\quad = 2 \int _{0}^{l}\rho _{0} \bigl(\ln (\theta + \delta ) \bigr)\,dy - \int _{0}^{l}\rho _{0} \bigl(\ln (\theta _{0} + \delta ) \bigr)\,dy+ \int _{0}^{l} \int _{0}^{t}\frac{\rho _{0}\theta }{(\theta +\delta )g}u_{y}\,dyd \tau \\ &\quad \leq 2 \int _{0}^{l}\rho _{0} \bigl((\theta + \delta ) \bigr)\,dy - \int _{0}^{l}\rho _{0} \bigl(\ln (\theta _{0} + \delta ) \bigr)\,dy+ \int _{0}^{l} \int _{0}^{t}\frac{\rho _{0}\theta }{(\theta +\delta )g}u_{y}\,dyd \tau \\ &\quad \leq 2 \int _{0}^{l}\rho _{0} \bigl((\theta + \delta ) \bigr)\,dy + \int _{0}^{l} \int _{0}^{t} \frac{\rho _{0}^{2}\theta ^{2}}{(\theta +\delta )g}\,dy\,d\tau + \frac{1}{2} \int _{0}^{l} \int _{0}^{t} \frac{u_{y}^{2}}{\theta +\delta )g}\,dy\,d\tau + C \\ &\quad \leq \frac{1}{2} \int _{0}^{l} \int _{0}^{t} \frac{u_{y}^{2}}{(\theta +\delta )g}\,dy\,d\tau + C\bigl( \Vert \rho \Vert _{\infty },m_{0}, E_{0}\bigr) (1+t). \end{aligned}$$
Thus we get
$$\begin{aligned} \int _{0}^{l}\rho _{0} \bigl(\ln (\theta + \delta ) \bigr) + \int _{0}^{l} \int _{0}^{t}\kappa \biggl( \frac{\theta _{y}^{2}}{(\theta + \delta )^{2}g} \biggr)\,dy\,d\tau \leq C(1+t). \end{aligned}$$
(2.36)
From Proposition 2, inequality \((i)_{4}\), it holds that
$$\begin{aligned} \Vert \theta \Vert _{L^{\infty }}\leq C(1+ t)^{2}. \end{aligned}$$
(2.37)
This completes Lemma 4. □
Combining all these lemmas, we have the following proposition.
Proposition 3
Given \(T\in (0,\infty )\). It holds that
$$\begin{aligned} &\sup_{0\leq t\leq T} \biggl\Vert \biggl(\sqrt{\rho _{0}}u^{2},\sqrt{ \rho _{0}}\theta , \biggl\vert \frac{d_{y}}{g^{\frac{3}{4}}} \biggr\vert ^{2} \biggr) \biggr\Vert _{2}^{2} + \int _{0}^{T} \biggl( \Vert \sqrt{\rho _{0}} \theta \Vert _{\infty }^{2} + \frac{1}{\sqrt{\rho _{0}}} \biggl\vert \frac{ d_{y}}{g} \biggr\vert ^{2}\| _{\infty }^{2} \biggr)\,dt \\ &\qquad {} + \int _{0}^{T} \biggl\Vert \biggl(\partial _{y} \theta ,u\partial _{y} u, \frac{1}{\sqrt{g}} \biggl( \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr)_{y}, \mu \frac{\partial _{y} u}{\sqrt{ g}}, \frac{1}{\sqrt{g}} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr) \biggr\Vert _{2}^{2}\,dt \\ &\quad \leq C \end{aligned}$$
(2.38)
and
$$ \sup_{0\leq t\leq T} \Vert g \Vert _{\infty }^{2}+ \int _{0}^{T} \Vert \theta \Vert _{\infty }^{2}\,dt\leq C $$
(2.39)
for a positive constant C depending only on the initial data R, \(c_{v}\), μ, κ, \(m_{1}\), \(\mathcal{N}_{1}\), and T, where \(m_{1}\) and \(\mathcal{N}_{1}\) are the numbers in Proposition 1and Proposition 2, respectively.
Proof
Summing with (2.28), one obtains
$$\begin{aligned} &\frac{d}{dt} \biggl( \Vert \sqrt{\rho _{0}}\mathcal{E} \Vert _{2}^{2} + \frac{C_{1}}{\mu } \bigl\Vert \sqrt{\rho _{0}}u^{2} \bigr\Vert _{2}^{2} + \biggl\Vert \biggl( \frac{d_{y}}{g^{\frac{3}{4}}} \biggr)^{2} \biggr\Vert _{2}^{2} \biggr) \\ &\qquad {} + \int _{0}^{t} \Vert \sqrt{\rho _{0}} \theta \Vert _{\infty }^{2}\,d\tau + \int _{0}^{t} \biggl\Vert \frac{1}{\sqrt{\rho _{0}}} \biggl\vert \frac{ d_{y}}{g} \biggr\vert ^{2} \biggr\Vert _{\infty }^{2}(t)\,d\tau \\ &\qquad {} +5C_{1} \biggl\Vert \frac{u\partial _{y} u}{\sqrt{g}} \biggr\Vert _{2}^{2} +\frac{\kappa c_{v}}{2} \biggl\Vert \frac{\partial _{y}\theta }{\sqrt{g}} \biggr\Vert _{2}^{2} + \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{d_{y}}{g} \cdot \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr) \biggr\Vert _{2}^{2} \\ &\quad \leq C \biggl(1+\frac{R}{\mu } \int _{0}^{t} \Vert \rho _{0} \theta \Vert _{\infty }\,d\tau \biggr) \end{aligned}$$
for any \(t\in (0,T)\), where C is a positive constant depending only on R, \(c_{v}\), μ, κ, \(m_{1}\), \(\mathcal{N}_{1}\), and T. Applying the Gronwall inequality to the above inequality, one gets
$$\begin{aligned} &\sup_{0\leq t\leq T} \biggl\Vert \biggl(\sqrt{\rho _{0}}u^{2},\sqrt{ \rho _{0}}\theta , \biggl( \frac{d_{y}}{g^{\frac{3}{4}}} \biggr)^{2} \biggr) \biggr\Vert _{2}^{2} + \int _{0}^{T} \biggl( \Vert \sqrt{\rho _{0}} \theta \Vert _{\infty }^{2} + \frac{1}{\sqrt{\rho _{0}}} \biggl\vert \frac{ d_{y}}{g} \biggr\vert ^{2}\| _{\infty }^{2} \biggr)\,dt \\ &\qquad {} + \int _{0}^{T} \biggl\Vert \biggl(\partial _{y} \theta ,u \partial _{y} u, \frac{1}{\sqrt{g}} \biggl( \frac{d_{y}}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr) \biggr) \biggr\Vert _{2}^{2}\,dt \\ &\quad \leq C \end{aligned}$$
(2.40)
for a positive constant C depending only on the initial data R, \(c_{v}\), μ, κ, \(m_{1}\), \(\mathcal{N}_{1}\), and T. Thus the desired estimates
$$ \sup_{0\leq t\leq T} \Vert g \Vert _{\infty }^{2}+ \int _{0}^{T} \Vert \theta \Vert _{\infty }^{2}\,dt\leq C $$
follow from (2.40) by applying Proposition 1 and (i) of Proposition 2. □
2.2 A priori \(H^{1}\) estimates
This section is devoted to the a priori \(H^{1}\) type estimates on \((g, u, d_{y},\theta )\). Precisely, we will carry out the a priori \(L^{\infty }(0,T; H^{1})\cap L^{2}(0,T; H^{2})\) estimate on u, a priori \(L^{\infty }(0,T; H^{1})\cap L^{2}(0,T; H^{2})\) estimate on \(d_{y}\), and the \(L^{\infty }(0,T; H^{1})\) estimate on g; however, due to the presence of the term \(\frac{\mu }{g}(\partial _{y} u)^{2}\) and \(g \vert \frac{1}{g} (\frac{d_{y}}{g} )_{y}+ \frac{1}{g^{2}}\vert d_{y}\vert ^{2}d \vert ^{2}\) on the right-hand side of the equation for θ (1.8)4, one cannot get the desired a priori \(H^{1}\) estimate of θ independent of the lower bound of the density without appealing to the higher than \(H^{1}\) energy estimates. Before going to prove the \(H^{1}\) bound for velocity, we first give the following estimates of director field, because the velocity field estimates strongly depend on the director field.
Lemma 5
There exists a positive constant \(C>0\) such that, for any \(t\in [0,T]\),
$$\begin{aligned}& \int _{0}^{l} \biggl(\frac{1}{g} \biggl\vert \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\vert ^{2} + g \vert d_{t} \vert ^{2} \biggr)\,dy + \int _{0}^{T} \int _{0}^{l} \frac{ \vert d_{yt} \vert ^{2}}{g}\,dy\,dt \leq C, \end{aligned}$$
(2.41)
$$\begin{aligned}& \max_{y\in [0,1]} \biggl\vert \frac{d_{y}}{g} \biggr\vert \leq C. \end{aligned}$$
(2.42)
Proof
Differentiating (1.8)3 with respect to time, taking the dot product by \((\frac{d_{y}}{g} )_{y}\), and integrating over interval \(y\in [0,l]\), we obtain
$$ \int _{0}^{l}d_{tt}\cdot \biggl( \frac{d_{y}}{g} \biggr)_{y}\,dy = \int _{0}^{l} \frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{yt}\cdot \biggl( \frac{d_{y}}{g} \biggr)_{y}\,dy + \int _{0}^{l} \biggl( \frac{ \vert d_{y} \vert ^{2}d}{g^{2}} \biggr)_{t}\cdot \biggl( \frac{d_{y}}{g} \biggr)_{y}\,dy. $$
(2.43)
The term on the left-hand side of (2.43) can be written with the aid of (1.8)1 and constraints \(\vert d\vert ^{2}=1\) as follows:
$$\begin{aligned} \int _{0}^{l}d_{tt}\cdot \biggl( \frac{d_{y}}{g} \biggr)_{y}\,dy ={}& \frac{d}{dt} \int _{0}^{l}d_{t}\cdot \biggl( \frac{d_{y}}{g} \biggr)_{y}\,dy - \int _{0}^{l}d_{t}\cdot \biggl( \frac{d_{y}}{g} \biggr)_{yt}\,dy \\ ={}&\frac{d}{dt} \int _{0}^{l} \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y}+ \frac{ \vert d_{y} \vert ^{2}d}{g^{2}} \biggr)\cdot \biggl(\frac{d_{y}}{g} \biggr)_{y}\,dy \\ &{} + \int _{0}^{l}d_{yt}\cdot \biggl( \frac{d_{y}}{g} \biggr)_{t}\,dy \\ ={}&\frac{d}{dt} \int _{0}^{l}\frac{1}{g} \biggl\vert \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr\vert ^{2}\,dy - \frac{d}{dt} \int _{0}^{l} \frac{1}{g^{3}} \vert d_{y} \vert ^{4}\,dy + \int _{0}^{l}\frac{1}{g} \vert d_{yt} \vert ^{2}\,dy \\ &{} - \int _{0}^{l}\frac{1}{g^{2}}(u_{y}d_{yt} \cdot d_{y})\,dy. \end{aligned}$$
(2.44)
The first term on the right-hand side of (2.43) can be written as
$$\begin{aligned} \int _{0}^{l}\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{yt} \cdot \biggl(\frac{d_{y}}{g} \biggr)_{y}\,dy=\frac{1}{2}\frac{d}{dt} \int _{0}^{l} \frac{1}{g} \biggl\vert \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\vert ^{2}\,dy + \int _{0}^{l}\frac{u_{y}}{2g^{2}} \biggl\vert \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\vert ^{2}\,dy, \end{aligned}$$
(2.45)
where we have used equation (1.8)1. Inserting (2.44) and (2.45) into (2.43), we get
$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \int _{0}^{l}\frac{1}{g} \biggl\vert \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr\vert ^{2}\,dy + \int _{0}^{l} \frac{ \vert d_{yt} \vert ^{2}}{g}\,dy \\ &\quad = \frac{d}{dt} \int _{0}^{l}\frac{1}{g^{3}} \vert d_{y} \vert ^{4}\,dy + \int _{0}^{l}\frac{1}{g^{2}}u_{y}( d_{yt}\cdot d_{y})\,dy + \int _{0}^{l} \frac{u_{y}}{2g^{2}} \biggl\vert \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\vert ^{2}\,dy \\ &\qquad {} + \int _{0}^{l} \biggl(\frac{ \vert d_{y} \vert ^{2}d}{g^{2}} \biggr)_{t} \cdot \biggl(\frac{d_{y}}{g} \biggr)_{y}\,dy \\ &\quad =: \frac{d}{dt} \int _{0}^{l}\frac{1}{g^{3}} \vert d_{y} \vert ^{4}\,dy+ \sum_{i=1}^{3}Q_{i}. \end{aligned}$$
(2.46)
Next the estimate of each term is given as follows: The term \(Q_{1}\) is estimated as
(2.47)
Now the estimate on the last term \(Q_{3}\) is given as
$$\begin{aligned} Q_{3} ={}& \int _{0}^{l} \biggl(\frac{ \vert d_{y} \vert ^{2}d}{g^{2}} \biggr)_{t}\cdot \biggl(\frac{d_{y}}{g} \biggr)_{y}\,dy \\ \leq{}& \int _{0}^{l} \biggl(- \frac{2 \vert d_{y} \vert ^{2}\,dg_{t}}{g^{3}} + \frac{2(d_{y}\cdot d_{yt})d}{g^{2}} + \frac{ \vert d_{y} \vert ^{2}d_{t}}{g^{2}} \biggr) \cdot \biggl( \frac{1}{g}d_{y} \biggr)_{y}\,dy \\ \leq {}&C \biggl( \int _{0}^{l}\frac{ \vert d_{y} \vert ^{4}}{g^{4}} \vert u_{y} \vert \,dy + \int _{0}^{l}\frac{1}{g^{3}} \vert d_{y} \vert ^{3} \vert d_{yt} \vert \,dy \biggr) \\ &{} + C \int _{0}^{l} \frac{ \vert d_{y} \vert ^{2}}{g^{2}} \vert \sqrt{g}d_{t} \vert \biggl\vert \frac{1}{\sqrt{g}} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr\vert \,dy \\ \leq{} &C \biggl(\max_{y\in [0,1]}\frac{ \vert d_{y} \vert ^{2}}{g^{2}} \biggr)^{\frac{3}{2}} \biggl\Vert \frac{u_{y}}{\sqrt{g}} \biggr\Vert _{L^{2}} \biggl\Vert \frac{d_{y}}{\sqrt{g}} \biggr\Vert _{L^{2}} + C\max_{y\in [0,1]} \biggl(\frac{ \vert d_{y} \vert ^{2}}{g^{2}} \biggr) \biggl\Vert \frac{d_{y}}{\sqrt{g}} \biggr\Vert _{L^{2}} \biggl\Vert \frac{d_{yt}}{\sqrt{g}} \biggr\Vert _{L^{2}} \\ &{} + C\max_{y\in [0,1]} \biggl( \frac{ \vert d_{y} \vert ^{2}}{g^{2}} \biggr) \Vert \sqrt{g}d_{t} \Vert _{L^{2}} \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{L^{2}} \\ \leq{}& \delta \biggl\Vert \frac{d_{yt}}{\sqrt{g}} \biggr\Vert _{L^{2}}^{2} + C \biggl( \biggl\Vert \frac{u_{y}}{\sqrt{g}} \biggr\Vert _{L^{2}}^{2}+ \Vert \sqrt{g}d_{t} \Vert _{L^{2}}^{2}+1 \biggr) \biggl\Vert \frac{1}{\sqrt{g}} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{L^{2}}^{2}. \end{aligned}$$
(2.48)
By using the Hölder inequality and embedding \(W^{1,1}(0,l) \hookrightarrow L^{\infty }(0,l)\), \(Q_{2}\) is estimated as
$$\begin{aligned} Q_{2} &\leq \biggl( \int _{0}^{l} \biggl\vert \frac{u_{y}}{\sqrt{g}} \biggr\vert ^{2}\,dy \biggr)^{\frac{1}{2}} \biggl( \int _{0}^{l} \biggl\vert \frac{1}{\sqrt{g}} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\vert ^{2}\,dy \biggr)^{\frac{1}{2}} \biggl(\max_{y\in [0,1]} \biggl\vert \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\vert ^{2} \biggr)^{\frac{1}{2}} \\ &\leq \biggl\Vert \frac{1}{\sqrt{g}}u_{y} \biggr\Vert _{L^{2}} \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{L^{2}} \biggl(\max _{y\in [0,1]} \biggl\vert \frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr\vert ^{2} \biggr)^{\frac{1}{2}}. \end{aligned}$$
(2.49)
By using the Sobolev embedding and equations (1.8)3, it follows that
$$\begin{aligned} & \biggl(\max_{y\in [0,1]} \biggl\vert \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\vert ^{2} \biggr)^{\frac{1}{2}} \end{aligned}$$
(2.50)
$$\begin{aligned} &\quad \leq \biggl\Vert \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{L^{2}} + \biggl( \int _{0}^{l} \biggl\vert \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \cdot \biggl(\frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\vert \,dy \biggr)^{\frac{1}{2}} \\ &\quad \leq \biggl\Vert \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{L^{2}} + \biggl( \int _{0}^{l} \biggl\vert \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \cdot \biggl(d_{yt}- \biggl(\frac{1}{g^{2}} \vert d_{y} \vert ^{2}d\biggr)_{y} \biggr) \biggr\vert \,dy \biggr)^{\frac{1}{2}} \\ &\quad \leq \biggl\Vert \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{L^{2}} + \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{L^{2}}^{\frac{1}{2}} \biggl\Vert \frac{1}{\sqrt{g}}{d_{yt}} \biggr\Vert _{L^{2}}^{\frac{1}{2}} \\ &\qquad {} + \biggl( \int _{0}^{l} \biggl\vert \frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y}\cdot \biggl(2\frac{d_{y}}{g} \cdot \biggl( \frac{ d_{y}}{g} \biggr)_{y}d + \frac{1}{g^{2}} \vert d_{y} \vert ^{2}d_{y} \biggr) \biggr\vert \,dy \biggr)^{\frac{1}{2}} \\ &\quad \leq \biggl\Vert \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{L^{2}} + \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{L^{2}}^{\frac{1}{2}} \biggl\Vert \frac{1}{\sqrt{g}}{d_{yt}} \biggr\Vert _{L^{2}}^{\frac{1}{2}} \\ &\qquad {} + C \biggl( \int _{0}^{l} \biggl\vert \frac{d_{y}}{g} \frac{ \vert d_{y} \vert ^{2}}{g^{2}}\cdot \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\vert \,dy \biggr)^{\frac{1}{2}} \\ &\quad \leq \biggl\Vert \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{L^{2}} + \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{L^{2}}^{\frac{1}{2}} \biggl\Vert \frac{1}{\sqrt{g}}{d_{yt}} \biggr\Vert _{L^{2}}^{\frac{1}{2}} \\ &\qquad {} + C\max_{y\in [0,1]} \biggl( \frac{ \vert d_{y} \vert ^{2}}{g^{2}} \biggr) \biggl\Vert \frac{d_{y}}{\sqrt{g}} \biggr\Vert _{L^{2}}^{\frac{1}{2}} \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{L^{2}}^{\frac{1}{2}}, \end{aligned}$$
(2.51)
whereas in the fourth inequality the following fact is used: \(\frac{1}{g} (\frac{d_{y}}{g} )_{y}\cdot d + \frac{\vert {d}_{y}\vert ^{2}}{g^{2}}=0\). Plugging (2.50) into (2.49), using Young’s inequality, we get
$$\begin{aligned} Q_{2} &\leq \biggl\Vert \frac{u_{y}}{\sqrt{g}} \biggr\Vert _{L^{2}} \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{L^{2}}^{2} + \biggl\Vert \frac{u_{y}}{\sqrt{g}} \biggr\Vert _{L^{2}} \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{L^{2}}^{ \frac{3}{2}} \biggl\Vert \frac{d_{yt}}{\sqrt{g}} \biggr\Vert _{L^{2}}^{ \frac{1}{2}} \\ &\leq \delta \biggl\Vert \frac{d_{yt}}{\sqrt{g}} \biggr\Vert _{L^{2}}^{2} + C \biggl( \biggl\Vert \frac{u_{y}}{\sqrt{g}} \biggr\Vert _{L^{2}}^{2}+1 \biggr) \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{L^{2}}^{2}. \end{aligned}$$
(2.52)
Plugging the estimates \(Q_{1}\), \(Q_{2}\), \(Q_{3}\) into (2.46) and choosing δ small enough, it holds
$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \int _{0}^{l}\frac{1}{g} \biggl\vert \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr\vert ^{2}\,dy + \int _{0}^{l} \frac{ \vert d_{yt} \vert ^{2}}{g}\,dy \\ &\quad = \frac{d}{dt} \int _{0}^{l}\frac{ \vert d_{y} \vert ^{4}}{g^{3}}\,dy + C \biggl( \biggl\Vert \frac{u_{y}}{\sqrt{g}} \biggr\Vert _{L^{2}}^{2}+ \Vert \sqrt{g}d_{t} \Vert _{L^{2}}^{2}+1 \biggr) \biggl\Vert \frac{1}{\sqrt{g}} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{L^{2}}^{2}. \end{aligned}$$
(2.53)
Integrating (2.53) with respect to time over the interval \([0,t]\), the resultant reads as
$$\begin{aligned} &\frac{1}{2} \int _{0}^{l}\frac{1}{g} \biggl\vert \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr\vert ^{2}\,dy + \int _{0}^{t} \int _{0}^{l} \frac{ \vert d_{yt} \vert ^{2}}{g}\,dy\,ds \\ &\quad \leq C \biggl(1 + \int _{0}^{l}\frac{ \vert d_{y} \vert ^{4}}{g^{3}}\,dy+ \int _{0}^{t} \biggl( \biggl\Vert \frac{u_{y}}{\sqrt{g}} \biggr\Vert _{L^{2}}^{2}+ \Vert \sqrt{g}d_{t} \Vert _{L^{2}}^{2}+1 \biggr) \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{L^{2}}^{2}\,ds \biggr) \\ &\quad \leq C \biggl(1 + \max_{y\in [0,1]} \biggl( \frac{ \vert d_{y} \vert ^{2}}{g^{2}} \biggr) \biggl\Vert \frac{d_{y}}{\sqrt{g}} \biggr\Vert _{L^{2}}^{2} \biggr) \\ &\qquad {} + C \int _{0}^{t} \biggl( \biggl\Vert \frac{u_{y}}{\sqrt{g}} \biggr\Vert _{L^{2}}^{2}+ \biggl\Vert \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{L^{2}}^{2}+ 1 \biggr) \biggl\Vert \frac{1}{\sqrt{g}} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{L^{2}}^{2}\,ds, \end{aligned}$$
(2.54)
where we have used the fact that
$$\begin{aligned} \int _{0}^{l}g \vert d_{t} \vert ^{2}\,dy \leq C \biggl(1 + \int _{0}^{l} \frac{1}{g} \biggl\vert \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\vert ^{2}\,dy \biggr). \end{aligned}$$
(2.55)
Using the estimates in hand, energy conservation (2.2), and the Gronwall inequality, it follows
$$\begin{aligned} \frac{1}{2} \int _{0}^{l}\frac{1}{g} \biggl\vert \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\vert ^{2}\,dy + \int _{0}^{t} \int _{0}^{l} \frac{ \vert d_{yt} \vert ^{2}}{g}\,dy\,ds \leq C, \end{aligned}$$
(2.56)
which deduces that \(\int _{0}^{l}g\vert d_{t}\vert ^{2}\,dy \leq C\). Similarly, by the Sobolev embedding \(W^{1,1}([0,l]) \hookrightarrow L^{\infty }([0,l])\), inequality (2.42) is estimated. The proof of Lemma 5 is completed. □
Define the effective viscous flux F as
$$ F:=\mu \frac{\partial _{y}u}{g}-p -\frac{1}{2} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2}=\mu \frac{\partial _{y}u}{g}- R \frac{\rho _{0}}{g}\theta - \frac{1}{2} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2}. $$
Then, one can derive from (1.8) that
$$\begin{aligned} &\partial _{t}F-\frac{\mu }{g}\partial _{y} \biggl( \frac{\partial _{y}F}{\rho _{0}} \biggr) \\ &\quad =-\frac{\kappa R}{c_{v}g}\partial _{y} \biggl( \frac{\partial _{y}\theta }{g} \biggr)-\biggl(\frac{R}{c_{v}}+1\biggr) \frac{\partial _{y}u}{g}F - \frac{d_{y}}{g}\cdot \biggl( \frac{d_{y}}{g} \biggr)_{t} - \frac{R}{c_{v}} \vert d_{t} \vert ^{2} - \frac{R}{2c_{v}} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \frac{u_{y}}{g} \\ &\qquad {}-\frac{1}{2} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2}\frac{u_{y}}{g} \\ &\quad =-\frac{\kappa R}{c_{v}g}\partial _{y} \biggl( \frac{\partial _{y}\theta }{g} \biggr)-\biggl(\frac{R}{c_{v}}+1\biggr) \frac{\partial _{y}u}{g}F - \frac{d_{y}}{g}\cdot \frac{d_{yt}}{g} - \frac{R}{c_{v}} \vert d_{t} \vert ^{2} \\ &\qquad {}-\biggl(\frac{R}{2c_{v}}+\frac{1}{2}\biggr) \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \frac{u_{y}}{g}. \end{aligned}$$
(2.57)
Moreover, by equation (1.8)2, one has \(\partial _{y}F=\rho _{0}\partial _{t}u\), from which, recalling the boundary conditions, we have
$$ \partial _{y}F(0,t)=\partial _{y}F(l,t)=0, \quad t\in (0,\infty ). $$
We have the a priori \(L^{2}\) estimates on F stated in the following.
Proposition 4
Given \(T\in (0,\infty )\). It holds that
$$ \sup_{0\leq t\leq T} \Vert F \Vert _{2}^{2}+ \int _{0}^{T} \biggl\Vert \frac{\partial _{y} F}{\sqrt{\rho _{0}}} \biggr\Vert _{2}^{2}\,dt\leq C $$
for a positive constant C depending only on R, \(c_{v}\), μ, κ, \(m_{1}\), \(\mathcal{N}_{1}\), N, and T, where
$$ N:= \bigl\Vert \sqrt{\rho _{0}}u_{0}^{2} \bigr\Vert _{2}+ \Vert \sqrt{\rho _{0}}\theta _{0} \Vert _{2}+ \bigl\Vert u_{0}' \bigr\Vert _{2} + \bigl\Vert d_{0}' \bigr\Vert _{2}, $$
and \(m_{1}\) and \(\mathcal{N}_{1}\) are the numbers in Proposition 1and Proposition 2, respectively.
Proof
Multiplying equation (2.57) by gF, integrating the resultant over \((0,l)\), and recalling \(\partial _{y}F|_{y=0,l}=0\), one gets from integration by parts that
$$\begin{aligned} & \int _{0}^{l}\partial _{t}FgF\,dy+\mu \int _{0}^{l} \biggl\vert \frac{\partial _{y}F}{\sqrt{\rho _{0}}} \biggr\vert ^{2}\,dy \\ &\quad =\kappa \frac{R}{c_{v}} \int _{0}^{l} \frac{\partial _{y}\theta \partial _{y}F}{g}\,dy- \biggl( \frac{R}{c_{v}}-1 \biggr) \int _{0}^{l}\partial _{y}uF^{2}\,dy \\ &\qquad - \int _{0}^{l}\frac{d_{y}}{g}\cdot d_{yt}F\,dy - \frac{R}{c_{v}} \int _{0}^{l} \vert d_{t} \vert ^{2}gF\,dy -\biggl(\frac{R}{2c_{v}}-1\biggr) \int _{0}^{l} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2}u_{y}F\,dy. \end{aligned}$$
Using (1.8)1, one has
$$\begin{aligned} \int _{0}^{l}\partial _{t}FgF\,dy =& \frac{1}{2}\frac{d}{dt} \int _{0}^{l} gF^{2}\,dy -\frac{1}{2} \int _{0}^{l} \partial _{t}gF^{2}\,dy \\ =&\frac{1}{2}\frac{d}{dt} \int _{0}^{l} gF^{2}\,dy-\frac{1}{2} \int _{0}^{l} \partial _{y}uF^{2}\,dy. \end{aligned}$$
Therefore, it follows from the Hölder, Young, and Gagliardo–Nirenberg inequalities and Corollary 3 that
$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \int _{0}^{l}gF^{2}\,dy+ \mu \int _{0}^{l} \biggl\vert \frac{\partial _{y}F}{\sqrt{\rho _{0}}} \biggr\vert ^{2}\,dy \\ &\quad =\kappa \biggl(\frac{R}{c_{v}}-1 \biggr) \int _{0}^{l} \frac{\partial _{y}\theta \partial _{y}F}{g}\,dy+ \biggl( \frac{3}{2}- \frac{R}{c_{v}} \biggr) \int _{0}^{l}\partial _{y}uF^{2}\,dy \\ &\qquad {}- \int _{0}^{l}\frac{d_{y}}{g}\cdot d_{yt}F\,dy - \frac{R}{c_{v}} \int _{0}^{l} \vert d_{t} \vert ^{2}gF\,dy +\biggl(1-\frac{R}{2c_{v}}\biggr) \int _{0}^{l} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2}u_{y}F\,dy \\ &\quad =:\sum_{i=1}^{5}K_{i}. \end{aligned}$$
Now we estimate each term as follows:
$$\begin{aligned} \sum_{i=1}^{5}K_{i}\leq{}& \kappa \biggl\vert \frac{R}{c_{v}}-1 \biggr\vert \sqrt{\bar{\rho }} \biggl\Vert \frac{\partial _{y}F}{\sqrt{\rho _{0}}} \biggr\Vert _{2} \biggl\Vert \frac{\partial _{y}\theta }{\sqrt{g}} \biggr\Vert _{2}+ \biggl(\frac{1}{2}- \frac{R}{c_{v}} \biggr) \Vert \partial _{y}u \Vert _{2} \Vert F \Vert _{2} \Vert F \Vert _{\infty } \\ &{}+\delta \biggl( \biggl\Vert \frac{d_{yt}}{\sqrt{g}} \biggr\Vert ^{2} + \frac{1}{\sqrt{g}} \biggl\Vert \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert ^{2}+ 1 \biggr)+ C( \delta ) \biggl[ \biggl\Vert \bigg\vert \frac{d_{y}}{g} \bigg\vert ^{2} \biggr\Vert _{\infty } \\ &{}+ \biggl(1-\frac{R}{2c_{v}}\biggr) \Vert u_{y} \Vert _{2}^{2} + \frac{R}{c_{v}} \Vert d_{t} \Vert _{2}^{2} \biggr] \Vert F \Vert _{2}^{2} \\ \leq{}& C \biggl( \biggl\Vert \frac{\partial _{y}F}{\sqrt{\rho _{0}}} \biggr\Vert _{2} \Vert \partial _{y}\theta \Vert _{2}+ \Vert \partial _{y}u \Vert _{2} \Vert F \Vert _{2}^{ \frac{3}{2}} \bigl( \Vert F \Vert _{2} + \Vert \partial _{y}F \Vert _{2}\bigr)^{\frac{1}{2}} \biggr) \\ &{}+\delta \biggl( \biggl\Vert \frac{d_{yt}}{\sqrt{g}} \biggr\Vert ^{2} + \frac{1}{\sqrt{g}} \biggl\Vert \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert ^{2}+ 1 \biggr) \\ &{}+ C(\delta ) \biggl( \biggl\Vert \bigg\vert \frac{d_{y}}{g} \bigg\vert ^{2} \biggr\Vert _{\infty } + \biggl(1-\frac{R}{2c_{v}}\biggr) \Vert u_{y} \Vert _{2}^{2} + \frac{R}{c_{v}} \Vert d_{t} \Vert _{2}^{2} \biggr) \Vert F \Vert _{2}^{2} \\ \leq{}& \frac{\mu }{2} \biggl\Vert \frac{\partial _{y}F}{\sqrt{\rho _{0}}} \biggr\Vert _{2}^{2} +C \biggl[ \biggl(1+ \biggl\Vert \bigg\vert \frac{d_{y}}{g} \bigg\vert ^{2} \biggr\Vert _{\infty } + \Vert d_{t} \Vert _{2}^{2}+ \Vert \partial _{y}u \Vert _{2}^{2} \biggr) \Vert F \Vert _{2}^{2}+ \Vert \partial _{y} \theta \Vert _{2}^{2} \\ &{}+ \biggl( \biggl\Vert \frac{d_{yt}}{\sqrt{g}} \biggr\Vert ^{2}+ \frac{1}{\sqrt{g}} \biggl\Vert \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert ^{2}+ 1 \biggr) \biggr], \end{aligned}$$
that is,
$$\begin{aligned} &\frac{d}{dt} \Vert \sqrt{g}F \Vert _{2}^{2}+\mu \biggl\Vert \frac{\partial _{y}F}{\sqrt{\rho _{0}}} \biggr\Vert _{2}^{2} \\ &\quad \leq C \biggl[ \biggl(1+ \biggl\Vert \bigg\vert \frac{d_{y}}{g} \bigg\vert ^{2} \biggr\Vert _{\infty } + \Vert d_{t} \Vert _{2}^{2}+ \Vert \partial _{y}u \Vert _{2}^{2} \biggr) \Vert F \Vert _{2}^{2} \\ &\qquad {} + \Vert \partial _{y}\theta \Vert _{2}^{2} + \biggl( \biggl\Vert \frac{d_{yt}}{\sqrt{g}} \biggr\Vert ^{2}+ \frac{1}{\sqrt{g}} \biggl\Vert \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{2}^{2}+ 1 \biggr) \biggr] \end{aligned}$$
(2.58)
for any \(t\in (0,T)\), where C is a positive constant depending only on R, \(c_{v}\), μ, κ, \(m_{1}\). Applying the Gronwall inequality to (2.58) and using Corollary 3, the conclusion follows. □
Based on Proposition 4 and Corollary 3, we can obtain the desired \(H^{1}\) type estimates on g, u, \(\frac{1}{g} (\frac{d_{y}}{g} )_{y}\), stated as follows.
Proposition 5
Given \(T\in (0,\infty )\). It holds that
$$\begin{aligned} &\sup_{0\leq t\leq T} \biggl( \Vert \partial _{y}g \Vert _{2}^{2}+ \Vert \partial _{y}u \Vert _{2}^{2} + \biggl\Vert \frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{2}^{2} + \Vert d_{t} \Vert _{2}^{2} \biggr) \\ &\qquad {}+ \int _{0}^{T} \bigl( \Vert \sqrt{\rho _{0}}\partial _{t}u \Vert _{2}^{2}+ \bigl\Vert \partial _{y}^{2}u \bigr\Vert _{2}^{2} \bigr)\,dt + \int _{0}^{T} \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} + d \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr)_{y} \biggr\Vert ^{2}\,dt \\ &\quad \leq C \end{aligned}$$
(2.59)
for a positive constant C depending only on R, \(c_{v}\), μ, κ, \(m_{1}\), \(\mathcal{N}_{1}\), \(\mathcal{N}_{2}\), and T, where \(m_{1}\), \(\mathcal{N}_{1}\), and \(\mathcal{N}_{2}\) are the numbers in Propositions 1, 2, and 4, respectively.
Proof
$$ \underbrace{\sup_{0\leq t\leq T} \biggl( \Vert \partial _{y}u \Vert _{2}^{2} + \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{2}^{2}+ \Vert d_{t} \Vert _{2}^{2} \biggr)}_{I_{1}} + \underbrace{ \int _{0}^{T} \Vert \sqrt{\rho _{0}} \partial _{t}u \Vert _{2}^{2} + \biggl\Vert \frac{d_{yt}}{\sqrt{g}} \biggr\Vert _{2}^{2}\,dt}_{I_{2}} \leq C. $$
The estimates of velocity term in \(I_{1}\) and \(I_{2}\) are straightforward from a priori estimates in hand and by the definition of F, noticing that \(\rho _{0}\partial _{t}u=\partial _{y}F\). Note that, by the Sobolev embedding inequality, it follows from Proposition 4 that
$$ \int _{0}^{T} \Vert F \Vert _{\infty }^{2}\,dt\leq C,\qquad \int _{0}^{l} \Vert F \Vert _{H^{1}}^{2}\,dt \leq C $$
(2.60)
for a positive constant C depending only on R, \(c_{v}\), μ, κ, \(m_{1}\), \(\mathcal{N}_{1}\), \(\mathcal{N}_{2}\), and T.
However, the a priori estimates on terms \(\|\partial _{y}g\|_{2}^{2}\) and \(\int _{0}^{T}\|\partial _{y}^{2}u\|_{2}^{2}\,dt\) need to be computed.
Rewrite (1.8)1 in terms of F as
$$ \partial _{t}g=\frac{1}{\mu } \biggl(gF+R\rho _{0} \theta + \frac{g}{2} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr). $$
Differentiating the above equations in y, multiplying the resultant by \(\partial _{y}g\), and integrating over \((0,l)\), it follows from the Hölder and Young inequalities that
$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \Vert \partial _{y}g \Vert _{2}^{2}={}&\frac{1}{\mu } \int _{0}^{l} \bigl(F \vert \partial _{y} g \vert ^{2}+\partial _{y}Fg\partial _{y}g+R\bigl(\rho _{0}' \theta +\rho _{0} \partial _{y} \theta \bigr)\partial _{y}g \bigr)\,dy \\ &{}+\frac{1}{\mu } \int _{0}^{l} \biggl[d_{y}\cdot \biggl( \frac{d_{y}}{g} \biggr)_{y}\partial _{y}g+ \frac{1}{2} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \vert \partial _{y}g \vert ^{2} \biggr]\,dy=:\mathcal{T}. \end{aligned}$$
Now we compute \(\mathcal{T}\) as
$$\begin{aligned} \mathcal{T}\leq{}& \frac{1}{\mu } \bigl( \Vert F \Vert _{\infty } \Vert \partial _{y}g \Vert _{2}^{2}+ \Vert g \Vert _{\infty } \Vert \partial _{y}F \Vert _{2} \Vert \partial _{y}g \Vert _{2}+R\bigl( \Vert \theta \Vert _{\infty } \bigl\Vert \rho _{0}' \bigr\Vert _{2}+\bar{\rho } \Vert \partial _{y}\theta \Vert _{2}\bigr) \Vert \partial _{y}g \Vert _{2} \bigr) \\ &{} + C\max_{y\in [0,1]} \biggl\vert \frac{d_{y}}{g} \biggr\vert \biggl\Vert \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{2} \Vert \partial _{y}g \Vert _{2} + \frac{1}{2} \biggl\Vert \frac{d_{y}}{g} \biggr\Vert _{\infty }^{2} \Vert \partial _{y}g \Vert _{2}^{2} \\ \leq {}&C \biggl( \Vert F \Vert _{\infty }^{2}+ \biggl\Vert \frac{d_{y}}{g} \biggr\Vert _{\infty }^{2}+ 1 \biggr) \Vert \partial _{y}g \Vert _{2}^{2} \\ &{}+C\bigl( \Vert g \Vert _{\infty }^{2} \Vert \partial _{y} F \Vert _{2}^{2}+ \bigl\Vert \rho _{0}' \bigr\Vert _{2}^{2} \Vert \theta \Vert _{\infty }^{2}+\bar{\rho }^{2} \Vert \partial _{y}\theta \Vert _{2}^{2}\bigr)+ C \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{2}^{2} \end{aligned}$$
for a positive constant C depending only on R and μ. Applying the Gronwall inequality, it follows from (2.60), Corollary 3, and Proposition 4 that
$$ \sup_{0\leq t\leq T} \Vert \partial _{y}g \Vert _{2}^{2}\leq C $$
(2.61)
for a positive constant C depending only on R, \(c_{v}\), μ, κ, \(m_{1}\), \(\mathcal{N}_{1}\), \(\mathcal{N}_{2}\), and T.
Noticing that \(\partial _{y}u=\frac{1}{\mu } (gF+R\rho _{0}\theta + \frac{g}{2} \vert \frac{d_{y}}{g} \vert ^{2} )\), one has
$$\begin{aligned} \partial _{y}^{2}u=\frac{1}{\mu } \biggl(\partial _{y}gF+g\partial _{y}F+R \rho _{0}' \theta +R\rho _{0}\partial _{y}\theta +d_{y}\cdot \biggl( \frac{d_{y}}{g} \biggr)_{y} + \frac{1}{2} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2}g_{y} \biggr), \end{aligned}$$
thus by the Hölder inequality, (2.60), (2.61), it follows from Corollary 3 and Proposition 4 that
$$\begin{aligned} &\int _{0}^{T} \bigl\Vert \partial _{y}^{2}u \bigr\Vert _{2}^{2}\,dt \\ &\quad \leq C \int _{0}^{T}\bigl( \Vert \partial _{y}g \Vert _{2}^{2} \Vert F \Vert _{\infty }^{2}+ \Vert g \Vert _{\infty }^{2} \Vert \partial _{y}F \Vert _{2}^{2}+ \bigl\Vert \rho _{0}' \bigr\Vert _{2}^{2} \Vert \theta \Vert _{\infty }^{2}+ \Vert \partial _{y}\theta \Vert _{2}^{2}\bigr) \\ &\qquad {}+ C \biggl[ \biggl(\max_{t\in [0,T]} \biggl(\max _{y \in [0,1]} \biggl\vert \frac{d_{y}}{g} \biggr\vert \biggr)^{2}+ 1 \biggr) \int _{0}^{T} \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{2}^{2}\,dt + \int _{0}^{T} \Vert u_{y} \Vert _{2}^{2}\,dt \biggr] \\ &\quad \leq C \int _{0}^{T} \biggl( \Vert F \Vert _{\infty }^{2}+ \Vert g \Vert _{\infty }^{2} \Vert \partial _{y}F \Vert _{2}^{2}+ \Vert \theta \Vert _{\infty }^{2}+ \Vert \partial _{y} \theta \Vert _{2}^{2} + \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{2}^{2}+ \Vert u_{y} \Vert _{2}^{2} \biggr)\,dt \\ &\quad \leq C \end{aligned}$$
for a positive constant C depending only on R, \(c_{v}\), μ, κ, \(m_{1}\), \(\mathcal{N}_{1}\), \(\mathcal{N}_{2}\), and T, proving the conclusion. □
We summarize the estimates obtained in this subsection as follows.
Corollary 1
Given \(T\in (0,\infty )\). It holds that
$$\begin{aligned} &\sup_{0\leq t\leq T} \biggl\Vert \biggl(F,\partial _{y} g, \partial _{y} u, \frac{1}{\sqrt{g}} \biggl(\frac{d_{y}}{g} \biggr)_{y},\sqrt{g}d_{t} \biggr) \biggr\Vert ^{2} \\ &\quad {} + \int _{0}^{T} \biggl\Vert \biggl( \frac{\partial _{y}F}{\sqrt{\rho _{0}}},\partial _{y}^{2} u, \sqrt{ \rho _{0}}\partial _{t} u , \frac{d_{yt}}{\sqrt{g}} \biggr) \biggr\Vert _{2}^{2}\,dt \leq C \end{aligned}$$
for a positive constant C depending only on R, \(c_{v}\), μ, κ, \(m_{1}\), \(\mathcal{N}_{1}\), \(\mathcal{N}_{2}\), and T, where \(m_{1}\), \(\mathcal{N}_{1}\), and \(\mathcal{N}_{2}\) are the numbers in Propositions 1, 2, and 4, respectively.
2.3 A priori \(H^{2}\) estimates
This subsection is devoted to the a prior \(H^{2}\) estimates on \((g, u, d_{y}, \theta )\). As will be shown in this subsection, one can get the desired a priori \(L^{\infty }(0,T; H^{2})\) estimate of θ without using the a priori \(L^{\infty }(0,T; H^{1})\) bound of it.
As a preparation, we first give some estimates on \(\|\partial _{y}\theta \|_{2}\) and \(\|\partial _{t}\theta \|_{\infty }\) in terms of \(\|\sqrt{\rho _{0}}\partial _{t}\theta \|_{2}\) and \(\|\partial _{y}\partial _{t}\theta \|_{2}\), and \(\|\partial _{t}u\|_{\infty }\), in terms of \(\|\sqrt{\rho _{0}}\partial _{t} u\|_{2}\) and \(\|\partial _{y}\partial _{t} u\|_{2}\), which will be used later in higher order a priori estimates.
Proposition 6
Given \(T\in (0,\infty )\); (i) It holds that
$$\begin{aligned} \Vert \partial _{y}\theta \Vert _{2}^{2}\leq C\bigl(1+ \Vert \sqrt{\rho _{0}} \partial _{t} \theta \Vert _{2}\bigr) \end{aligned}$$
for a positive constant C depending only on R, \(c_{v}\), μ, κ, \(m_{1}\), \(\mathcal{N}_{1}\), \(\mathcal{N}_{2}\), and T, where \(m_{1}\), \(\mathcal{N}_{1}\), and \(\mathcal{N}_{2}\) are the numbers in Propositions 1, 2, and 4, respectively.
(ii) It holds that
$$\begin{aligned}& \Vert \partial _{t}\theta \Vert _{\infty } \leq \sqrt{ \frac{2}{\Omega _{0}\bar{\rho }}} \Vert \sqrt{\rho _{0}}\partial _{t} \theta \Vert _{2} +\sqrt{l} \biggl\Vert \frac{\partial _{y}\partial _{t}\theta }{ \sqrt{g}} \biggr\Vert _{2}, \\& \Vert \partial _{t}u \Vert _{\infty } \leq \sqrt{ \frac{2}{\Omega _{0}\bar{\rho }}} \Vert \sqrt{\rho _{0}}\partial _{t} u \Vert _{2} +\sqrt{l} \biggl\Vert \frac{\partial _{y}\partial _{t}u}{ \sqrt{g}} \biggr\Vert _{2}, \end{aligned}$$
where \(\Omega _{0}\) is the number in Proposition 2.
Proof
(i) Multiplying (2.86) by θ, integrating the resultant over \((0,l)\), and integrating by parts, it follows from the Hölder inequality that
$$\begin{aligned} &\kappa \int _{0}^{l} \biggl\vert \frac{\partial _{y}\theta }{\sqrt{g}} \biggr\vert ^{2}\,dy \\ &\quad = \int _{0}^{l} \biggl(\partial _{y}uF + \frac{1}{2} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \partial _{y}uF + g \biggl\vert \frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} + \frac{1}{g^{2}} \vert d_{y} \vert ^{2}d\biggr\vert ^{2} -c_{v}\rho _{0}\partial _{t}\theta \biggr) \theta \,dy \\ &\quad \leq \biggl( \Vert \partial _{y}u \Vert _{2} \Vert F \Vert _{2} + \Vert \partial _{y}u \Vert _{2} \Vert F \Vert _{2} \biggl\Vert \frac{d_{y}}{g} \biggr\Vert _{\infty }^{2} + \Vert \sqrt{g}d_{t} \Vert _{2}^{2} \biggr) \Vert \theta \Vert _{\infty }+c_{v} \Vert \sqrt{\rho _{0}} \partial _{t}\theta \Vert _{2} \Vert \sqrt{\rho _{0}}\theta \Vert _{2}, \end{aligned}$$
from which, by Corollaries 3–1 and (i) of Proposition 2, we have that
$$\begin{aligned} \Vert \partial _{y}\theta \Vert _{2}^{2} \leq& C \int _{0}^{l} \biggl\vert \frac{\partial _{y}\theta }{\sqrt{g}} \biggr\vert ^{2}\,dy \leq C\bigl( \Vert \theta \Vert _{\infty }+ \Vert \sqrt{\rho _{0}} \partial _{t} \theta \Vert _{2}\bigr) \\ \leq &C\bigl( \Vert \partial _{y}\theta \Vert _{2}+1+ \Vert \sqrt{\rho _{0}} \partial _{t} \theta \Vert _{2}\bigr)\leq \frac{1}{2} \Vert \partial _{y}\theta \Vert _{2}^{2} +C\bigl(1+ \Vert \sqrt{\rho _{0}} \partial _{t}\theta \Vert _{2}\bigr), \end{aligned}$$
and thus,
$$ \Vert \partial _{y}\theta \Vert _{2}^{2}\leq C\bigl(1+ \Vert \sqrt{\rho _{0}} \partial _{t} \theta \Vert _{2}\bigr) $$
for a positive constant C depending only on R, \(c_{v}\), μ, κ, \(m_{1}\), \(\mathcal{N}_{1}\), \(\mathcal{N}_{2}\), and T.
(ii) Recall that \(\Omega _{0}:= \{ y\in (0,l) |\rho _{0}(y)\geq \frac{\bar{\rho }}{2} \} \) and \(|\Omega _{0}|>0\). Noticing
$$ \partial _{t}\theta (y,t)=\frac{1}{ \vert \Omega _{0} \vert } \int _{ \Omega _{0}} \partial _{t}\theta (z,t)\,dz + \frac{1}{ \vert \Omega _{0} \vert } \int _{\Omega _{0}} \int _{z}^{y} \partial _{y}\partial _{t}\theta (\xi ,t)\,d\xi \,dz, $$
it follows from the Hölder inequality that
$$\begin{aligned} \bigl\vert \partial _{t}\theta (y,t) \bigr\vert \leq & \frac{1}{ \vert \Omega _{0} \vert } \biggl\vert \int _{\Omega _{0}} \frac{\sqrt{\rho _{0}}\partial _{t}\theta }{\sqrt{\rho _{0}}}\,dz \biggr\vert + \int _{0}^{l} \bigl\vert \partial _{y} \partial _{t}\theta (\xi ,t) \bigr\vert \,d\xi \\ \leq &\sqrt{\frac{2}{ \vert \Omega _{0} \vert \bar{\rho }}} \Vert \sqrt{\rho _{0}} \partial _{t} \theta \Vert _{2} + \biggl( \int _{0}^{l} \biggl\vert \frac{\partial _{y}\partial _{t}\theta }{\sqrt{g}} \biggr\vert ^{2}\,d\xi \biggr)^{\frac{1}{2}} \biggl( \int _{0}^{l} g\,d\xi \biggr)^{\frac{1}{2}} \\ =&\sqrt{\frac{2}{ \vert \Omega _{0} \vert \bar{\rho }}} \Vert \sqrt{\rho _{0}} \partial _{t} \theta \Vert _{2} +\sqrt{l} \biggl\Vert \frac{\partial _{y}\partial _{t}\theta }{ \sqrt{g}} \biggr\Vert _{2}, \end{aligned}$$
which implies
$$ \Vert \partial _{t}\theta \Vert _{\infty } \leq \sqrt{ \frac{2}{ \vert \Omega _{0} \vert \bar{\rho }}} \Vert \sqrt{\rho _{0}}\partial _{t} \theta \Vert _{2} +\sqrt{l} \biggl\Vert \frac{\partial _{y}\partial _{t}\theta }{ \sqrt{g}} \biggr\Vert _{2}. $$
In the same way as above, the same conclusion holds for \(\partial _{t}u\). □
Proposition 7
Given \(T\in (0,\infty )\). It holds that
$$ \sup_{0\leq t\leq T} \biggl\Vert \biggl(\sqrt{\rho _{0}} \partial _{t} \theta , \frac{\partial _{y}F}{\sqrt{\rho _{0}}} \biggr) \biggr\Vert _{2}^{2} + \int _{0}^{T} \bigl\Vert (\partial _{t}F,\partial _{y}\partial _{t}\theta ) \bigr\Vert _{2}^{2}\,dt\leq C\bigl( \Vert h_{1} \Vert _{2}+ \Vert h_{2} \Vert _{2}\bigr) $$
for a positive constant C depending only on R, \(c_{v}\), μ, κ, \(m_{1}\), \(\mathcal{N}_{1}\), \(\mathcal{N}_{2}\), and T, where
$$ h_{1}:= \frac{\mu u_{0}^{\prime \prime }-R(\rho _{0}\theta _{0})^{\prime } + ( \vert d_{0}^{\prime } \vert ^{2})^{\prime }}{\sqrt{\rho _{0}}},\qquad h_{2}:=\frac{1}{\sqrt{\rho _{0}}} \bigl[\mu \bigl(u_{0}^{\prime }\bigr)^{2}+ \kappa \theta _{0}^{\prime \prime } + \bigl( \vert d_{0t} \vert ^{2}\bigr)^{\prime } -Ru_{0}'\rho _{0}\theta _{0} \bigr], $$
and \(m_{1}\), \(\mathcal{N}_{1}\), and \(\mathcal{N}_{2}\) are the numbers in Propositions 1, 2, and 4, respectively.
Proof
Rewrite (1.8)4 as
$$ c_{v}\rho _{0}\partial _{t} \theta -\kappa \partial _{y} \biggl( \frac{\partial _{y} \theta }{g} \biggr)= \partial _{y}uF + \frac{1}{2} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2}\partial _{y}u + g \biggl\vert \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} + \frac{1}{g^{2}} \vert d_{y} \vert ^{2}d \biggr\vert ^{2}, $$
(2.62)
or equivalently,
$$\begin{aligned} c_{v}\rho _{0}\partial _{t}\theta -\kappa \partial _{y} \biggl( \frac{\partial _{y} \theta }{g} \biggr)={}&\frac{1}{\mu } \biggl(gF+R\rho _{0}\theta + \frac{1}{2g} \vert d_{y} \vert ^{2} \biggr)F \\ &{}+ \biggl(g \biggl\vert \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y}+ \frac{1}{g^{2}} \vert d_{y} \vert ^{2}d \biggr\vert ^{2} + \frac{1}{g^{2}} \vert d_{y} \vert ^{2} u_{y} \biggr), \end{aligned}$$
(2.63)
from which, differentiating in t and using (1.8)1, one has
$$\begin{aligned} &c_{v}\rho _{0}\partial _{t}^{2} \theta -\kappa \partial _{y} \biggl( \frac{\partial _{y}\partial _{t}\theta }{g} - \frac{\partial _{y}u\partial _{y}\theta }{g^{2}} \biggr) \\ &\quad =\frac{1}{\mu }\bigl(\partial _{y}uF^{2}+2gF\partial _{t}F\bigr) + \frac{R\rho _{0}}{\mu }(\partial _{t} \theta F+ \theta \partial _{t}F) + \frac{1}{2\mu } \biggl(\partial _{t} \biggl( \frac{ \vert d_{y} \vert ^{2}}{g} \biggr) F+ \frac{ \vert d_{y} \vert ^{2}}{g}\partial _{t}F \biggr) \\ &\qquad {}+ \biggl(g \biggl\vert \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y}+ \frac{1}{g^{2}} \vert d_{y} \vert ^{2}d \biggr\vert ^{2} + \frac{1}{g^{2}} \vert d_{y} \vert ^{2} u_{y} \biggr)_{t} \\ &\quad =\frac{\partial _{y}u}{\mu }F^{2}+\frac{1}{\mu } \biggl(2gF+R\rho _{0} \theta + \frac{1}{2}\frac{ \vert d_{y} \vert ^{2}}{g} \biggr) \partial _{t}F+ \biggl(\frac{R}{\mu }\rho _{0}\partial _{t}\theta + \frac{1}{2\mu } \partial _{t} \biggl( \frac{ \vert d_{y} \vert ^{2}}{g} \biggr) \biggr)F \\ &\qquad {}+ \biggl(g \biggl\vert \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y}+ \frac{1}{g^{2}} \vert d_{y} \vert ^{2}d \biggr\vert ^{2} + \frac{1}{g^{2}} \vert d_{y} \vert ^{2} u_{y} \biggr)_{t}. \end{aligned}$$
Multiplying the above equation by \(\partial _{t}\theta \), integrating the resultant over \((0,l)\), one gets from integration by parts that
$$\begin{aligned} &\frac{c_{v}}{2}\frac{d}{dt} \int _{0}^{l}\rho _{0} \vert \partial _{t} \theta \vert ^{2}\,dy +\kappa \int _{0}^{l} \biggl\vert \frac{\partial _{y}\partial _{t}\theta }{\sqrt{g}} \biggr\vert ^{2}\,dy \\ &\quad =\kappa \int _{0}^{l}\frac{\partial _{y}u\partial _{y}\theta }{g^{2}} \partial _{y}\partial _{t}\theta \,dy + \int _{0}^{l} \biggl(\frac{1}{\mu }R \rho _{0}F(\partial _{t}\theta ) + \frac{1}{2\mu }\partial _{t} \biggl( \frac{ \vert d_{y} \vert ^{2}}{g} \biggr)F \biggr)\partial _{t}\theta \,dy \\ &\qquad {}+\frac{1}{\mu } \int _{0}^{l} \biggl[ \biggl(2gF+R\rho _{0} \theta + \frac{1}{2}\frac{ \vert d_{y} \vert ^{2}}{g} \biggr)\partial _{t}F+ \partial _{y}uF^{2} \biggr]\partial _{t}\theta \,dy \\ &\qquad {}+ \int _{0}^{l} \biggl(g \biggl\vert \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y}+ \frac{1}{g^{2}} \vert d_{y} \vert ^{2}d \biggr\vert ^{2} + \frac{1}{g^{2}} \vert d_{y} \vert ^{2} u_{y} \biggr)_{t}\partial _{t} \theta \,dy. \end{aligned}$$
(2.64)
The terms on the right-hand side of (2.64) are estimated as follows. By Corollary 3, it follows from the Young inequality and (i) of Proposition 6 that
$$\begin{aligned} \kappa \int _{0}^{l}\frac{\partial _{y}u\partial _{y}\theta }{g^{2}} \partial _{y}\partial _{t}\theta \,dy \leq &\frac{\kappa }{4} \biggl\Vert \frac{\partial _{y}\partial _{t}\theta }{\sqrt{g}} \biggr\Vert _{2}^{2}+C \Vert \partial _{y}u \Vert _{\infty }^{2} \Vert \partial _{y}\theta \Vert _{2}^{2} \\ \leq &\frac{\kappa }{4} \biggl\Vert \frac{\partial _{y}\partial _{t}\theta }{\sqrt{g}} \biggr\Vert _{2}^{2}+C \biggl( \Vert F \Vert _{\infty }^{2}+ \Vert \theta \Vert _{\infty }^{2} + \biggl\Vert \bigg\vert \frac{d_{y}}{g}\bigg\vert ^{2} \biggr\Vert _{\infty }^{2} \biggr) \Vert \partial _{y}\theta \Vert _{2}^{2} \\ \leq &\frac{\kappa }{4} \biggl\Vert \frac{\partial _{y}\partial _{t} \theta }{\sqrt{g}} \biggr\Vert _{2}^{2} \\ &{}+C \biggl( \Vert F \Vert _{\infty }^{2}+ \Vert \theta \Vert _{\infty }^{2} + \biggl\Vert \bigg\vert \frac{d_{y}}{g} \bigg\vert ^{2} \biggr\Vert _{\infty }^{2} \biggr) \bigl(1+ \Vert \sqrt{\rho _{0}} \partial _{t}\theta \Vert _{2}\bigr) \end{aligned}$$
for a positive constant C depending only on R, \(c_{v}\), μ, κ, \(m_{1}\), \(\mathcal{N}_{1}\), and T. By Corollary 3, Corollary 1, and (ii) of Proposition 6, it follows from the Hölder and Young inequalities that
$$\begin{aligned} &\frac{1}{\mu } \int _{0}^{l} \biggl[ \biggl(2gF+R\rho _{0} \theta + \frac{1}{2}\frac{ \vert d_{y} \vert ^{2}}{g} \biggr)\partial _{t}F + \partial _{y}uF^{2} \biggr]\partial _{t}\theta \,dy \\ &\quad \leq C \biggl[ \biggl( \Vert g \Vert _{\infty } \Vert F \Vert _{2}+ \Vert \sqrt{\rho _{0}}\theta \Vert _{2} + \biggl\Vert \frac{ d_{y}}{\sqrt{g}} \biggr\Vert _{2}^{2} \biggr) \Vert \partial _{t} F \Vert _{2}+ \Vert \partial _{y}u \Vert _{2} \Vert F \Vert _{2} \Vert F \Vert _{\infty } \biggr] \Vert \partial _{t} \theta \Vert _{\infty } \\ &\quad \leq C\bigl( \Vert \partial _{t}F \Vert _{2}+ \Vert F \Vert _{\infty }\bigr) \biggl( \Vert \sqrt{\rho _{0}} \partial _{t}\theta \Vert _{2}+ \biggl\Vert \frac{\partial _{y}\partial _{t}\theta }{\sqrt{g}} \biggr\Vert _{2} \biggr) \\ &\quad \leq \frac{\kappa }{4} \biggl\Vert \frac{\partial _{y}\partial _{t}\theta }{\sqrt{g}} \biggr\Vert _{2}^{2}+C \bigl( \Vert \sqrt{g}\partial _{t}F \Vert _{2}^{2}+ \Vert \sqrt{\rho _{0}}\partial _{t} \theta \Vert _{2}^{2}+ \Vert F \Vert _{\infty }^{2} \bigr) \end{aligned}$$
for a positive constant C depending only on R, \(c_{v}\), μ, κ, \(m_{1}\), \(\mathcal{N}_{1}\), \(\mathcal{N}_{2}\), and T. Before going to estimate the next term, we first compute
$$\begin{aligned} & \int _{0}^{l} \biggl(2g \biggl\vert \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y}+ \frac{1}{g^{2}} \vert d_{y} \vert ^{2}d \biggr\vert ^{2} + \frac{1}{g^{2}} \vert d_{y} \vert ^{2} u_{y} \biggr)_{t}\theta _{t}\,dy \\ &\quad = \int _{0}^{l} \biggl(2g \vert d_{t} \vert ^{2} + \frac{1}{g^{2}} \vert d_{y} \vert ^{2} u_{y} \biggr)_{t}\theta _{t}\,dy \\ &\quad = \int _{0}^{l} \biggl[4g d_{t}d_{tt} + 2 \vert d_{t} \vert ^{2}u_{y} + \frac{1}{g^{2}} \vert d_{y} \vert ^{2} u_{yt} + \frac{u_{y}}{g} \biggl( \frac{ \vert d_{y} \vert ^{2}}{g} \biggr)_{t} - \frac{(u_{y})^{2}}{g^{2}} \frac{ \vert d_{y} \vert ^{2}}{g} \biggr]\theta _{t}\,dy \\ &\quad =:\mathcal{U.} \end{aligned}$$
(2.65)
The right-hand side is estimated as follows:
$$\begin{aligned} \mathcal{U}\leq{}& \delta \bigl( \Vert \sqrt{g}d_{tt} \Vert ^{2}+ \Vert u_{yt} \Vert _{2}^{2} \bigr) + C \Vert \sqrt{g}d_{t} \Vert ^{2} \Vert \theta _{t} \Vert _{\infty }^{2} + C \biggl\Vert \frac{d_{y}}{\sqrt{g}} \biggr\Vert _{\infty }^{2} \Vert \theta _{t} \Vert _{ \infty }^{2} \\ &{}+ C \Vert u_{y} \Vert _{2}^{2}\| \bigg\vert \frac{d_{y}}{g} \bigg\vert ^{2} \| _{\infty }^{2} + \frac{1}{2} \Vert u_{y} \Vert _{2}^{2} \Vert \theta _{t} \Vert _{ \infty }^{2} \\ \leq{}& \delta \biggl( \Vert \sqrt{g}d_{tt} \Vert ^{2}+ \biggl[\partial _{t} \biggl(\frac{ \vert d_{y} \vert ^{2}}{g} \biggr) \biggr]^{2} + \Vert \sqrt{g} \partial _{t}G \Vert _{2}^{2} \biggr) + C \Vert u_{y} \Vert _{2}^{2}\| \bigg\vert \frac{d_{y}}{g} \bigg\vert ^{2}\| _{\infty }^{2} \\ &{}+ C \Vert \sqrt{g}d_{t} \Vert ^{2} \Vert \theta _{t} \Vert _{\infty }^{2}+ \frac{1}{2} \Vert u_{y} \Vert _{2}^{2} \Vert \theta _{t} \Vert _{\infty }^{2} + C \Vert \partial _{t}\theta \Vert _{ \infty }^{2}+ C \biggl\Vert \frac{d_{y}}{\sqrt{g}} \biggr\Vert _{\infty }^{2} \Vert \theta _{t} \Vert _{\infty }^{2} \\ \leq{}& \delta \biggl( \Vert \sqrt{g}d_{tt} \Vert ^{2}+ \biggl\Vert \partial _{t} \biggl(\frac{ \vert d_{y} \vert ^{2}}{g} \biggr) \biggr\Vert _{2}^{2} + \Vert \sqrt{g}\partial _{t}G \Vert _{2}^{2} \biggr) + C \Vert \theta _{t} \Vert _{\infty }^{2} \\ \leq{}& \delta \biggl( \Vert \sqrt{g}d_{tt} \Vert ^{2}+ C \biggl( \Vert u_{y} \Vert _{2}^{2} + \biggl\Vert \frac{d_{yt}}{\sqrt{g}} \biggr\Vert _{2}^{2} \biggr) + \Vert \sqrt{g} \partial _{t}G \Vert _{2}^{2} \biggr) + C \Vert \theta _{t} \Vert _{\infty }^{2} \\ \leq{}& \delta \bigl( \Vert \sqrt{g}d_{tt} \Vert ^{2} + \Vert \sqrt{g}\partial _{t}G \Vert _{2}^{2} \bigr) + C \Vert \theta _{t} \Vert _{\infty }^{2}. \end{aligned}$$
(2.66)
The second integral on the right-hand side of (2.64) is estimated as
$$\begin{aligned} & \int _{0}^{l} \biggl(\frac{1}{\mu }R\rho _{0}F(\partial _{t}\theta ) + \frac{1}{2\mu }\partial _{t} \biggl(\frac{ \vert d_{y} \vert ^{2}}{g} \biggr)F \biggr)\partial _{t}\theta \,dy \\ &\quad \leq \delta \biggl[\partial _{t} \biggl( \frac{ \vert d_{y} \vert ^{2}}{g} \biggr) \biggr]^{2} + C \Vert \partial _{t} \theta \Vert _{\infty }^{2} \\ &\quad \leq \delta \biggl( \Vert u_{y} \Vert _{2}^{2} + \biggl\Vert \frac{d_{yt}}{\sqrt{g}} \biggr\Vert _{2}^{2} \biggr)+ C \Vert \partial _{t} \theta \Vert _{\infty }^{2}, \end{aligned}$$
(2.67)
where we have used the Hölder, Young inequalities and mass conservation equation. Therefore, one obtains from (2.64) that
$$\begin{aligned} &c_{v}\frac{d}{dt} \Vert \sqrt{\rho _{0}}\partial _{t}\theta \Vert _{2}^{2} + \kappa \biggl\Vert \frac{\partial _{y}\partial _{t}\theta }{\sqrt{g}} \biggr\Vert _{2}^{2} \leq C_{3}\bigl( \Vert \sqrt{g}d_{tt} \Vert ^{2}+ \Vert \sqrt{g} \partial _{t}F \Vert _{2}^{2} \bigr) \\ &\quad + \biggl[ \biggl(1+ \Vert F \Vert _{\infty }^{2} + \Vert \theta \Vert _{\infty }^{2} + \biggl\Vert \bigg\vert \frac{d_{y}}{g} \bigg\vert ^{2} \biggr\Vert _{\infty }^{2} \biggr) \bigl( \Vert \sqrt{\rho _{0}}\partial _{t}\theta \Vert _{2}^{2}+1\bigr) \biggr] \end{aligned}$$
(2.68)
for a positive constant \(C_{3}\) depending only on R, \(c_{v}\), μ, κ, \(m_{1}\), \(\mathcal{N}_{1}\), \(\mathcal{N}_{2}\), and T. In order to control the above estimates, there is a need to get dissipation estimates on \((\|\sqrt{g}d_{tt}\|^{2}, \|\sqrt{g}\partial _{t}F\|_{2}^{2})\).
Next, we give the higher order derivative estimates of the director vector field d. The identity \(\frac{1}{g} (\frac{d_{y}}{g} )_{y}\cdot d + \frac{\vert {d}_{y}\vert ^{2}}{g^{2}}=0\), which is obtained from the director field equation with the help of constraint \(\vert d\vert ^{2}=1\), is frequently used in the proof of the lemma. This fact replaces the higher derivative with nonlinearity.
Lemma 6
For any \(0< t \leq T\), it holds that
$$\begin{aligned} & \int _{0}^{l} \biggl(\frac{1}{g} \biggl\vert \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\vert ^{2} + \frac{1}{g} \vert d_{yt} \vert ^{2} \biggr)\,dy + \int _{0}^{T} \int _{0}^{l}\frac{1}{g} \biggl\vert \biggl(\frac{d_{y}}{g} \biggr)_{yt} \biggr\vert ^{2}\,dy\,dt \leq C. \end{aligned}$$
(2.69)
Proof
In order to prove higher order derivative estimates on the director field, the derivative with respect to y variable is taken on both sides of director field equation (1.8)3 and the resultant is multiplied by \(\frac{1}{g}\), then it follows
$$ \frac{1}{g}d_{yt}= \frac{1}{g} \biggl( \frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} + \frac{1}{g} \biggl( \frac{d \vert {d}_{y} \vert ^{2}}{g^{2}} \biggr)_{y}. $$
(2.70)
Taking the dot product of (2.70) on both sides with \((\frac{1}{g} (\frac{d_{y}}{g} )_{y} )_{yt}\) and integrating over space variable, it follows that
$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \int _{0}^{l} \biggl(\frac{1}{g} \biggl\vert \biggl(\frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\vert ^{2} \biggr)\,dy \\ &\quad = - \int _{0}^{l}\frac{1}{g^{2}}g_{t} \frac{1}{g} \biggl\vert \biggl( \frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\vert ^{2}\,dy + 2 \int _{0}^{l} \frac{1}{g} \biggl( \frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y}\cdot \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{yt}\,dy \\ &\quad = - \int _{0}^{l}\frac{1}{g^{2}}u_{y} \frac{1}{g} \biggl\vert \biggl( \frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\vert ^{2}\,dy + 2 \int _{0}^{l}\frac{d_{yt}}{g}\cdot \biggl( \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr)_{yt}\,dy \\ &\qquad {} -2 \int _{0}^{l}\frac{1}{g} \biggl( \frac{ \vert d_{y} \vert ^{2}}{g^{2}}d\biggr)_{y} \cdot \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{yt}\,dy=: \sum _{i=1}^{3}R_{i}, \end{aligned}$$
(2.71)
where in the second equality we have used \(\frac{1}{g} (\frac{d_{y}}{g} )_{y}\cdot d + \frac{\vert {d}_{y}\vert ^{2}}{g^{2}}=0\). This fact replaces the higher derivative with nonlinearity. The key point is to control the higher derivative with lower one by using the constraint \(\vert d\vert ^{2}=1\). Now the terms \(R_{1}\), \(R_{2}\), \(R_{3}\) are estimated one by one.
The term \(R_{1}\) is estimated as
$$\begin{aligned} R_{1}&\leq \max_{y\in [0,1]} \biggl( \frac{u_{y}}{g} \biggr) \int _{0}^{l}\frac{1}{g} \biggl\vert \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\vert ^{2}\,dy \\ &\leq C \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\Vert _{L^{2}}^{2}. \end{aligned}$$
(2.72)
Before estimating the term \(R_{2}\), we first make use of (1.8)1 and compute the integral. The simplified form is written as
$$\begin{aligned} R_{2} ={}& 2 \int _{0}^{l}\frac{d_{yt}}{g}\cdot \biggl( \frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{yt}\,dy \\ ={}& 2 \int _{0}^{1} \biggl( \biggl(\frac{d_{y}}{g} \biggr)_{t}\cdot \biggl(\frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr)_{yt} - \frac{g_{t}d_{y}}{g^{2}}\cdot \biggl( \frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr)\,dy \\ ={}&-2 \int _{0}^{1} \biggl(\frac{d_{y}}{g} \biggr)_{yt}\cdot \biggl( \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr)_{t}\,dy + \int _{0}^{1} \frac{2u_{y}d_{y}}{g^{2}}\cdot \biggl( \frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y}\,dy \\ ={}& -2 \int _{0}^{l} \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{yt} \cdot \biggl(\frac{d_{y}}{g} \biggr)_{yt}- \frac{u_{y}}{g^{2}} \biggl( \frac{d_{y}}{g} \biggr)_{y}\cdot \biggl(\frac{d_{y}}{g} \biggr)_{yt} \biggr)\,dy \\ &{} + \int _{0}^{1}\frac{2u_{y}d_{y}}{g^{2}}\cdot \biggl( \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr)_{y}\,dy. \end{aligned}$$
(2.73)
Now we are in a position to estimate the term \(R_{2}\), where we will use the definition of effective viscous flux:
$$\begin{aligned} R_{2}\leq{}& -2 \biggl\Vert \frac{1}{\sqrt{g}} \biggl( \frac{d_{y}}{g} \biggr)_{yt} \biggr\Vert _{L^{2}}^{2} + 2\max_{y\in [0,1]} \biggl( \frac{u_{y}}{g} \biggr) \biggl\Vert \frac{1}{\sqrt{g}} \biggl( \frac{d_{y}}{g} \biggr)_{yt} \biggr\Vert _{L^{2}} \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{L^{2}} \\ &{} + 2\max_{y\in [0,1]} \biggl(\frac{u_{y}}{g} \biggr) \biggl\Vert \frac{d_{y}}{\sqrt{g}} \biggr\Vert _{L^{2}} \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\Vert _{L^{2}} \\ \leq{}& - \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{d_{y}}{g} \biggr)_{yt} \biggr\Vert _{L^{2}}^{2} \\ &{}+ C \biggl( \Vert F \Vert _{\infty }^{2}+ \Vert \theta \Vert _{\infty }^{2} + \biggl\Vert \frac{d_{y}}{g} \biggr\Vert _{\infty }^{4} \biggr) \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\Vert _{L^{2}}^{2} + C. \end{aligned}$$
(2.74)
It is hard to estimate the term \(R_{3}\), so there is the need to simplify the integral so that the required bounds can be obtained, thus we simplify it as
$$\begin{aligned} R_{3}={}& -2\frac{d}{dt} \int _{0}^{l}\frac{1}{g^{2}} \biggl( \frac{ \vert d_{y} \vert ^{2}}{g^{2}}d \biggr)_{y} \cdot \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y}\,dy \\ &{}-2 \int _{0}^{l}\frac{u_{y}}{g^{2}} \biggl( \frac{ \vert d_{y} \vert ^{2}}{g^{2}}d\biggr)_{y} \cdot \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y}\,dy \\ &{}+ 2 \int _{0}^{l}\frac{1}{g} \biggl( \frac{ \vert d_{y} \vert ^{2}}{g^{2}}d\biggr)_{yt}\cdot \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y}\,dy \\ =:{}& -2\frac{d}{dt} \int _{0}^{l}\frac{1}{g^{2}} \biggl( \frac{ \vert d_{y} \vert ^{2}}{g^{2}}d\biggr)_{y}\cdot \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y}\,dy + \sum _{i=1}^{2}R_{3i}. \end{aligned}$$
(2.75)
In order to compute the term \(R_{3}\), we estimate terms \(R_{31}\) and \(R_{32}\) one by one. The term \(R_{31}\) is controlled as follows:
$$\begin{aligned} R_{31}={}&-2 \int _{0}^{l}\frac{u_{y}}{g^{2}} \biggl( \frac{ 2d_{y}}{g} \cdot \biggl(\frac{d_{y}}{g} \biggr)_{y}d + \frac{ \vert d_{y} \vert ^{2}}{g^{2}}d_{y} \biggr) \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y}\,dy \\ \leq{}& 2\max_{y\in [0,1]} \biggl\vert \frac{u_{y}}{g} \biggr\vert \max_{y\in [0,1]} \biggl\vert \frac{d_{y}}{g} \biggr\vert \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{L^{2}} \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\Vert _{L^{2}} \\ &{} +2\max_{y\in [0,1]} \biggl\vert \frac{u_{y}}{g} \biggr\vert \max_{y\in [0,1]} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggl\Vert \frac{d_{y}}{\sqrt{g}} \biggr\Vert _{L^{2}} \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\Vert _{L^{2}} \\ \leq{}& C \biggl( \Vert F \Vert _{\infty }^{2}+ \Vert \theta \Vert _{\infty }^{2} + \biggl\Vert \frac{d_{y}}{g} \biggr\Vert _{\infty }^{4} \biggr) \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\Vert _{L^{2}}^{2} + C, \end{aligned}$$
(2.76)
where Lemma 1, Lemma 5, and the director field equation are used.
The term \(R_{32}\) cannot be estimated directly, so we first need to simply the integral, (1.8)1 and (1.8)3 are used frequently:
$$\begin{aligned} R_{32}={}&2 \int _{0}^{l}\frac{1}{g} \biggl( \frac{2d_{y}}{g}\cdot \biggl( \frac{d_{y}}{g} \biggr)_{y}d + \frac{ \vert d_{y} \vert ^{2}}{g^{2}}d_{y} \biggr)_{t} \cdot \biggl( \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr)_{y}\,dy \\ ={}& 2 \int _{0}^{l}\frac{1}{g} \biggl(2 \biggl( \frac{d_{y}}{g} \biggr)_{t} \cdot \biggl(\frac{d_{y}}{g} \biggr)_{y}d+\frac{2d_{y}}{g}\cdot \biggl(\frac{d_{y}}{g} \biggr)_{yt}d\biggr)\cdot \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y}\,dy \\ &{} +2 \int _{0}^{l}\frac{1}{g} \biggl( \frac{2d_{y}}{g}\cdot \biggl( \frac{d_{y}}{g} \biggr)_{y}d_{t} \biggr)\cdot \biggl(\frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr)_{y}\,dy \\ &{} +2 \int _{0}^{l}\frac{1}{g} \biggl( \frac{2d_{y}}{g}\cdot \biggl( \frac{d_{y}}{g} \biggr)_{t}d_{y}+ \frac{ \vert d_{y} \vert ^{2}}{g^{2}}d_{yt} \biggr) \cdot \biggl(\frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr)_{y}\,dy. \end{aligned}$$
Thus, after simplifying and collecting the terms, we have
$$\begin{aligned} R_{32}={}& 4 \int _{0}^{l}\frac{1}{g} \biggl( \biggl( \frac{d_{y}}{g} \biggr)_{t}\cdot \biggl(\frac{d_{y}}{g} \biggr)_{y}d \biggr)\cdot \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y}\,dy \\ & {}+4 \int _{0}^{l}\frac{1}{g} \biggl( \frac{d_{y}}{g}\cdot \biggl( \frac{d_{y}}{g} \biggr)_{yt}d \biggr) \cdot \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y}\,dy \\ & {}+4 \int _{0}^{l}\frac{1}{g} \biggl( \frac{d_{y}}{g}\cdot \biggl( \frac{d_{y}}{g} \biggr)_{y}d_{t} \biggr)\cdot \biggl(\frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr)_{y}\,dy \\ &{} +2 \int _{0}^{l}\frac{1}{g} \biggl( \frac{2d_{y}}{g}\cdot \biggl(- \frac{u_{y}d_{y}}{g^{2}} + \frac{d_{yt} }{g} \biggr)d_{y} \biggr)\cdot \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y}\,dy \\ &{} + 2 \int _{0}^{l}\frac{1}{g} \biggl( \frac{ \vert d_{y} \vert ^{2}}{g^{2}}d_{yt} \biggr)\cdot \biggl( \frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y}\,dy=: \sum _{i=1}^{5}R_{32i}. \end{aligned}$$
(2.77)
Next, we are in a position to estimate terms \(\sum_{i=1}^{5}R_{32i}\) one by one. The term \(R_{321}\) is estimated as follows:
$$\begin{aligned} R_{321}={}&4 \int _{0}^{l}\frac{1}{g} \biggl( \biggl(- \frac{u_{y}d_{y}}{g^{2}}+ \frac{d_{yt}}{g} \biggr)\cdot \biggl(\frac{d_{y}}{g} \biggr)_{y}d\biggr) \cdot \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y}\,dy \\ \leq{}& C \biggl( \Vert F \Vert _{\infty }^{2}+ \Vert \theta \Vert _{\infty }^{2} + \biggl\Vert \frac{d_{y}}{g} \biggr\Vert _{\infty }^{4} \biggr)\max_{y\in [0,1]} \biggl\vert \frac{d_{y}}{g} \biggr\vert \biggl\Vert \frac{1}{\sqrt{g}} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{L^{2}} \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\Vert _{L^{2}} \\ &{} +C\max_{y\in [0,1]} \biggl\vert \frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr\vert \biggl\Vert \frac{d_{yt}}{\sqrt{g}} \biggr\Vert _{L^{2}} \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\Vert _{L^{2}} \\ \leq{}& C \biggl( \Vert F \Vert _{\infty }^{2}+ \Vert \theta \Vert _{\infty }^{2} + \biggl\Vert \bigg\vert \frac{d_{y}}{g} \bigg\vert ^{2} \biggr\Vert _{\infty }^{2}+ \biggl\Vert \frac{d_{yt}}{\sqrt{g}} \biggr\Vert _{L^{2}}^{2} \biggr) \biggl(1+ \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\Vert _{L^{2}}^{2} \biggr). \end{aligned}$$
(2.78)
The term \(R_{322}\) of (2.3) is controlled as follows:
$$\begin{aligned} R_{322}&\leq C\max_{y\in [0,1]} \biggl(\frac{d_{y}}{g} \biggr) \biggl\Vert \frac{1}{\sqrt{g}} \biggl( \frac{d_{y}}{g} \biggr)_{yt} \biggr\Vert _{L^{2}} \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\Vert _{L^{2}} \\ &\leq \delta \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{d_{y}}{g} \biggr)_{yt} \biggr\Vert _{L^{2}}^{2}+ C \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\Vert _{L^{2}}^{2}, \end{aligned}$$
(2.79)
where the Holder and Young inequalities are used along with bounds of the director field in hand.
The term \(R_{324}\) is estimated as follows:
$$\begin{aligned} R_{324}={}&2 \int _{0}^{l}\frac{1}{g} \biggl( \frac{2d_{y}}{g}\cdot \biggl(- \frac{u_{y}d_{y}}{g^{2}} + \frac{d_{yt} }{g} \biggr)d_{y} \biggr)\cdot \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y}\,dy \\ \leq{}& C\max_{y\in [0,1]} \biggl\vert \frac{u_{y}}{g} \biggr\vert \max_{y\in [0,1]} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggl\Vert \frac{d_{y}}{\sqrt{g}} \biggr\Vert _{L^{2}} \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\Vert _{L^{2}} \\ &{} + \max_{y\in [0,1]} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggl\Vert \frac{d_{yt}}{\sqrt{g}} \biggr\Vert _{L^{2}} \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\Vert _{L^{2}} \\ \leq {}&C \biggl( \Vert F \Vert _{\infty }^{2}+ \Vert \theta \Vert _{\infty }^{2} + \biggl\Vert \bigg\vert \frac{d_{y}}{g} \bigg\vert ^{2} \biggr\Vert _{\infty }^{2}+ \biggl\Vert \frac{d_{yt}}{\sqrt{g}} \biggr\Vert _{L^{2}}^{2} \biggr) \\ &{} \times \biggl(1+ \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\Vert _{L^{2}}^{2} \biggr), \end{aligned}$$
(2.80)
where the definition of effective viscous flux and the estimates of director field and velocity are used. The terms \(R_{323}\) and \(R_{325}\) are estimated together as follows:
$$\begin{aligned} R_{323}+ R_{225}={}& \int _{0}^{1} \biggl(\frac{d_{y}}{g}\cdot \biggl( \frac{d_{y}}{g} \biggr)_{y}d_{t} + \frac{ \vert d_{y} \vert ^{2}}{g^{2}}d_{yt} \biggr) \cdot \frac{1}{g} \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y}\,dy \\ \leq{}& \max_{y\in [0,1]} \biggl\vert \frac{d_{y}}{g} \biggr\vert \max_{y\in [0,1]} \biggl\vert \frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr\vert \Vert \sqrt{g}d_{t} \Vert _{L^{2}} \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\Vert _{L^{2}} \\ &{} +\max_{y\in [0,1]} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggl\Vert \frac{d_{yt}}{\sqrt{g}} \biggr\Vert _{L^{2}} \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\Vert _{L^{2}} \\ \leq{}& C \biggl(1+ \biggl\Vert \frac{d_{yt}}{\sqrt{g}} \biggr\Vert _{L^{2}}^{2} \biggr) \biggl(1+ \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\Vert _{L^{2}}^{2} \biggr), \end{aligned}$$
(2.81)
where the Holder and Young inequalities are used along with bounds of the director field in hand. Combining all the above estimates with (2.71) and choosing δ small enough, the resultant is written as follows:
$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \int _{0}^{l} \biggl(\frac{1}{g} \biggl\vert \biggl(\frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\vert ^{2} \biggr)\,dy + \int _{0}^{l}\frac{1}{2g} \biggl\vert \biggl( \frac{d_{y}}{g} \biggr)_{yt} \biggr\vert ^{2}\,dy \\ &\quad \leq -2\frac{d}{dt} \int _{0}^{l}\frac{1}{g^{2}} \biggl( \frac{ \vert d_{y} \vert ^{2}}{g^{2}}d\biggr)_{y} \cdot \biggl( \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr)_{y}\,dy \\ &\qquad {}+ C \biggl( \Vert F \Vert _{\infty }^{2}+ \Vert \theta \Vert _{\infty }^{2} + \biggl\Vert \bigg\vert \frac{d_{y}}{g} \bigg\vert ^{2} \biggr\Vert _{\infty }^{2}+ \biggl\Vert \frac{d_{yt}}{\sqrt{g}} \biggr\Vert _{L^{2}}^{2} \biggr) \\ &\qquad {} \times \biggl(1+ \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\Vert _{L^{2}}^{2} \biggr). \end{aligned}$$
(2.82)
Integrating (2.82) over the time interval \([0,t]\), it is not hard to see that
$$\begin{aligned} &\frac{1}{2} \int _{0}^{l} \biggl(\frac{1}{g} \biggl\vert \biggl( \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\vert ^{2} \biggr)\,dy + \int _{0}^{t} \int _{0}^{l}\frac{1}{2g} \biggl\vert \biggl(\frac{d_{y}}{g} \biggr)_{yt} \biggr\vert ^{2}\,dy\,ds \\ &\quad \leq C \biggl( \Vert F \Vert _{\infty }^{2}+ \Vert \theta \Vert _{\infty }^{2} + \biggl\Vert \bigg\vert \frac{d_{y}}{g} \bigg\vert ^{2} \biggr\Vert _{\infty }^{2}+ \biggl\Vert \frac{d_{yt}}{\sqrt{g}} \biggr\Vert _{L^{2}}^{2} \biggr) \\ &\qquad {} \times \biggl(1+ \biggl\Vert \frac{1}{\sqrt{g}} \biggl(\frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\Vert _{L^{2}}^{2} \biggr). \end{aligned}$$
(2.83)
By using Gronwall’s inequality and (2.41), we have
$$\begin{aligned} \int _{0}^{l}\frac{1}{g} \biggl\vert \biggl(\frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr)_{y} \biggr\vert ^{2}\,dy + \int _{0}^{t} \int _{0}^{l}\frac{1}{g} \biggl\vert \biggl(\frac{d_{y}}{g} \biggr)_{yt} \biggr\vert ^{2}\,dy\,ds \leq C. \end{aligned}$$
(2.84)
Moreover, from (1.8)3 and inequality (2.84), we get
$$\begin{aligned} \int _{0}^{l}\frac{1}{g} \vert d_{yt} \vert ^{2}\,dy \leq C. \end{aligned}$$
(2.85)
Similarly, one can deduce from the director field equation that
$$\begin{aligned} \int _{0}^{T} \int _{0}^{l}g \vert d_{tt} \vert ^{2}\,dy\,dt \leq C. \end{aligned}$$
Combining (2.84) and (2.85), the proof of Lemma 6 is completed. □
Using temperature, one can rewrite (2.57) as
$$\begin{aligned}& -\frac{R}{c_{v}}\kappa \partial _{y} \biggl( \frac{\partial _{y} \theta }{g} \biggr) \\& \quad =- R\rho _{0}\partial _{t}\theta + \frac{R}{c_{v}} \biggl(\partial _{y}uF + \frac{1}{2} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2}\partial _{y}u + g \biggl\vert \frac{1}{g} \biggl( \frac{d_{y}}{g} \biggr)_{y} + \frac{1}{g^{2}} \vert d_{y} \vert ^{2}d\biggr\vert ^{2} \biggr), \\& \partial _{t}F-\frac{\mu }{g}\partial _{y} \biggl( \frac{\partial _{y}F}{\rho _{0}} \biggr) =-(R)\frac{\rho _{0}}{g} \partial _{t}\theta - \frac{1}{g}\partial _{y}uF - \frac{d_{y}}{g} \cdot \frac{d_{yt}}{g}. \end{aligned}$$
(2.86)
Multiplying the above equation by \(g\partial _{t}F\), integrating the resultant over \((0,l)\), and integrating by parts, it follows from the Hölder and Young inequalities, Corollary 3, and Corollary 1 that
$$\begin{aligned}& \frac{\mu }{2}\frac{d}{dt} \int _{0}^{l} \biggl\vert \frac{\partial _{y}F}{\sqrt{\rho _{0}}} \biggr\vert ^{2}\,dy + \int _{0}^{l} g \vert \partial _{t}F \vert ^{2}\,dy \\& \quad = -R \int _{0}^{l}\rho _{0}\partial _{t}\theta \partial _{t}F\,dy+ \int _{0}^{l} \partial _{y}uF\partial _{t}F\,dy - \int _{0}^{l} d_{y} \cdot \frac{d_{yt}}{g}\partial _{t}F\,dy \\& \quad \leq \frac{1}{2} \Vert \sqrt{g}\partial _{t}F \Vert _{2}^{2}+C \biggl( \Vert \sqrt{\rho _{0}} \partial _{t}\theta \Vert _{2}^{2}+ \Vert \partial _{y}v \Vert _{2}^{2} \Vert F \Vert _{\infty }^{2} + \biggl\Vert \bigg\vert \frac{d_{y}}{g} \bigg\vert ^{2} \biggr\Vert _{\infty } \biggl\Vert \frac{d_{yt}}{g} \biggr\Vert _{2}^{2} \biggr) \\& \quad \leq \frac{1}{2} \Vert \sqrt{g}\partial _{t}F \Vert _{2}^{2}+C \biggl( \Vert \sqrt{\rho _{0}} \partial _{t}\theta \Vert _{2}^{2}+ \Vert F \Vert _{\infty }^{2} + \biggl\Vert \bigg\vert \frac{d_{y}}{g} \bigg\vert ^{2} \biggr\Vert _{\infty } \biggr), \end{aligned}$$
thus
$$ \mu \frac{d}{dt} \biggl\Vert \frac{\partial _{y}F}{\sqrt{\rho _{0}}} \biggr\Vert _{2}^{2} + \Vert \sqrt{g}\partial _{t}F \Vert _{2}^{2}\leq C \biggl( \Vert \sqrt{\rho _{0}} \partial _{t}\theta \Vert _{2}^{2}+ \Vert F \Vert _{\infty }^{2} + \biggl\Vert \bigg\vert \frac{d_{y}}{g} \bigg\vert ^{2} \biggr\Vert _{\infty } \biggr) $$
(2.87)
for a positive constant C depending only on R, \(c_{v}\), μ, κ, \(m_{1}\), \(\mathcal{N}_{1}\), \(\mathcal{N}_{2}\), and T.
Multiplying (2.87) by \(2A_{3}\) and summing the resultant with Lemma 6 and (2.68), one obtains
$$\begin{aligned} &\frac{d}{dt} \biggl(c_{v} \Vert \sqrt{\rho _{0}} \partial _{t}\theta \Vert _{2}^{2} +2A_{3}\mu \biggl\Vert \frac{\partial _{y}F}{\sqrt{\rho _{0}}} \biggr\Vert _{2}^{2} \biggr) +\kappa \biggl\Vert \frac{\partial _{y}\partial _{t}\theta }{\sqrt{g}} \biggr\Vert _{2}^{2}+A_{3} \Vert \sqrt{g} \partial _{t}F \Vert _{2}^{2} \\ &\quad \leq C \biggl( \Vert \theta \Vert _{\infty }^{2}+ \Vert F \Vert _{\infty }^{2} + \biggl\Vert \bigg\vert \frac{d_{y}}{g} \bigg\vert ^{2} \biggr\Vert _{\infty } \biggr) \bigl( \Vert \sqrt{ \rho _{0}}\partial _{t}\theta \Vert _{2}^{2}+1 \bigr) \end{aligned}$$
for a positive constant C depending only on R, \(c_{v}\), μ, κ, \(m_{1}\), \(N_{1}\), \(N_{2}\), and T. Applying the Gronwall inequality to the above inequality, by Corollary 3, and using (2.60), the conclusion of Proposition 7 is completed. □
Proposition 8
Given \(T\in (0,\infty )\). It holds that
$$ \sup_{0\leq t\leq T} \bigl( \bigl\Vert \bigl(\partial _{y}^{2}g, \partial _{y}^{2}u, \partial _{y}\theta , \partial _{y}^{2}\theta \bigr) \bigr\Vert _{2}^{2}+ \Vert \theta \Vert _{\infty } \bigr) + \int _{0}^{T} \bigl\Vert \bigl(\partial _{y}^{3}u, \partial _{y}\partial _{t}u, \partial _{y}^{3}\theta \bigr) \bigr\Vert _{2}^{2}\leq C $$
for a positive constant C depending only on R, \(c_{v}\), μ, κ, \(m_{1}\), \(\mathcal{N}_{1}\), \(\mathcal{N}_{2}\), \(\mathcal{N}_{3}\), and T, where
$$ \mathcal{N}_{3}:= \bigl\Vert \rho _{0}'' \bigr\Vert _{2}+ \Vert h_{1} \Vert _{2}+ \Vert h_{2} \Vert _{2}, $$
and \(m_{1}\), \(\mathcal{N}_{1}\), and \(\mathcal{N}_{2}\) are the numbers in Propositions 1, 2, and 4, respectively.
Proof
Combining (i) of Proposition 6 and Proposition 7, we get
$$ \sup_{0\leq t\leq T} \Vert \partial _{y}\theta \Vert _{2}^{2}\leq C\sup_{0 \leq t\leq T}\bigl(1+ \Vert \sqrt{\rho _{0}}\partial _{t}\theta \Vert _{2}^{2}\bigr) \leq C\bigl(1+ \Vert h_{1} \Vert _{2}^{2}+ \Vert h_{2} \Vert _{2}^{2}\bigr) $$
(2.88)
and thus, by (i) of Proposition 2 and Corollary 3, we have that
$$ \sup_{0\leq t\leq T} \Vert \theta \Vert _{\infty }\leq C\sup _{0\leq t\leq T}\bigl( \Vert \partial _{y}\theta \Vert _{2}+1\bigr) \leq C\bigl(1+ \Vert h_{1} \Vert _{2}+ \Vert h_{2} \Vert _{2}\bigr) $$
(2.89)
for a positive constant C depending only on R, \(c_{v}\), μ, κ, \(m_{1}\), \(\mathcal{N}_{1}\), \(\mathcal{N}_{2}\), and T. Using (2.88)–(2.89), it follows from the Hölder inequality and Corollaries 1–3 that
$$\begin{aligned} \Vert \partial _{y}p \Vert _{2} =&R \biggl\Vert \frac{\rho _{0}'}{g}\theta + \frac{ \rho _{0}}{g}\partial _{y}\theta - \frac{\rho _{0}}{g^{2}}\partial _{y} g \theta \biggr\Vert _{2} \\ \leq &C\bigl( \bigl\Vert \rho _{0}' \bigr\Vert _{2} \Vert \theta \Vert _{\infty }+ \Vert \rho _{0} \Vert _{\infty } \Vert \partial _{y}\theta \Vert _{2}+ \Vert \rho _{0} \Vert _{\infty } \Vert \partial _{y}g \Vert _{2} \Vert \theta \Vert _{\infty }\bigr) \\ \leq &C\bigl(1+ \Vert h_{1} \Vert _{2}+ \Vert h_{2} \Vert _{2}\bigr) \end{aligned}$$
(2.90)
for a positive constant C depending only on R, \(c_{v}\), μ, κ, \(m_{1}\), \(\mathcal{N}_{1}\), \(\mathcal{N}_{2}\), and T.
Noticing that \(\partial _{y}u=\frac{1}{\mu } (gF+R\rho _{0}\theta + \frac{g}{2} \vert \frac{d_{y}}{g} \vert ^{2} )\) and using (1.8)1, one has
$$\begin{aligned}& \partial _{t}\partial _{y}u=\frac{1}{\mu } \biggl( \partial _{y}uF+g \partial _{t}F+R \rho _{0} \partial _{t}\theta + d_{y}\cdot \frac{d_{yt}}{g} - \frac{1}{2} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2}u_{y} \biggr), \\& \partial _{y}^{2}u=\frac{1}{\mu } \biggl(\partial _{y}gF+g\partial _{y}F+R \rho _{0}' \theta +R\rho _{0}\partial _{y}\theta + d_{y} \cdot \biggl( \frac{d_{y}}{g} \biggr)_{y} + \frac{1}{2} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2}g_{y} \biggr), \end{aligned}$$
and thus, by the Hölder and Sobolev embedding inequalities and using (2.88)–(2.89), it follows from Corollaries 3–1 and Proposition 7 that
$$\begin{aligned} & \int _{0}^{T} \Vert \partial _{t} \partial _{y}u \Vert _{2}^{2}\,dt \\ &\quad \leq C \int _{0}^{T} \biggl( \Vert \partial _{y}u \Vert _{2}^{2} \Vert F \Vert _{\infty }^{2}+ \Vert g \Vert _{\infty }^{2} \Vert \partial _{t}F \Vert _{2}^{2}+ \Vert \sqrt{\rho _{0}} \partial _{t} \theta \Vert _{2}^{2}+ \Vert d_{y} \Vert _{\infty }^{2} \biggl\Vert \frac{d_{yt}}{g} \biggr\Vert _{2}^{2} \biggr) \\ & \qquad {}+C \int _{0}^{T} \biggl( \biggl\Vert \bigg\vert \frac{d_{y}}{g} \bigg\vert ^{2} \biggr\Vert _{\infty }^{2} \Vert u_{y} \Vert _{2}^{2} \biggr)\,dt \\ &\quad \leq C \int _{0}^{T} \biggl( \Vert \partial _{y}u \Vert _{2}^{2} \biggl( \Vert F \Vert _{H^{1}}^{2}+ \biggl\Vert \bigg\vert \frac{d_{y}}{g} \bigg\vert ^{2} \biggr\Vert _{\infty }^{2} \biggr)+ \Vert g \Vert _{\infty }^{2} \Vert \partial _{t}F \Vert _{2}^{2}+ \Vert \sqrt{\rho _{0}} \partial _{t} \theta \Vert _{2}^{2} \biggr) \end{aligned}$$
(2.91)
$$\begin{aligned} &\qquad {} + C \int _{0}^{T} \biggl( \Vert d_{y} \Vert _{\infty }^{2} \biggl\Vert \frac{d_{yt}}{g} \biggr\Vert _{2}^{2} \biggr)\,dt \\ &\quad \leq C\bigl(1+ \Vert h_{1} \Vert _{2}^{2}+ \Vert h_{2} \Vert _{2}^{2}\bigr) \end{aligned}$$
(2.92)
and
$$\begin{aligned} \sup_{0\leq t\leq T} \bigl\Vert \partial _{y}^{2}u \bigr\Vert _{2}^{2}\leq{}& C \sup_{0 \leq t\leq T}\bigl( \Vert \partial _{y}g \Vert _{2}^{2} \Vert F \Vert _{\infty }^{2}+ \Vert g \Vert _{\infty }^{2} \Vert \partial _{y}F \Vert _{2}^{2}+ \bigl\Vert \rho _{0}' \bigr\Vert _{2}^{2} \Vert \theta \Vert _{\infty }^{2}+ \Vert \partial _{y}\theta \Vert _{2}^{2}\bigr) \\ \leq {}&C\sup_{0\leq t\leq T} \biggl[ \Vert \partial _{y}g \Vert _{2}^{2} \biggl( \Vert F \Vert _{H^{1}}^{2}+ \biggl\Vert \bigg\vert \frac{d_{y}}{g} \bigg\vert ^{2} \biggr\Vert _{\infty }^{2} \biggr)+ \Vert g \Vert _{\infty }^{2} \Vert \partial _{y}F \Vert _{2}^{2}+ \Vert \theta \Vert _{\infty }^{2} \\ & {}+ \Vert \partial _{y}\theta \Vert _{2}^{2} + C \Vert d_{y} \Vert _{\infty }^{2} \biggl\Vert \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{2}^{2} \biggr] \\ \leq{}& C\bigl(1+ \Vert h_{1} \Vert _{2}^{2}+ \Vert h_{2} \Vert _{2}^{2}\bigr) \end{aligned}$$
(2.93)
for a positive constant C depending only on R, \(c_{v}\), μ, κ, \(m_{1}\), \(\mathcal{N}_{1}\), \(\mathcal{N}_{2}\), and T. In the above inequality the constraint condition \(\vert d\vert ^{2}=1\) is used, respectively.
Using (2.86), we have
$$\begin{aligned} \partial _{y}^{2}\theta ={}&g\partial _{y} \biggl( \frac{\partial _{y}\theta }{g} \biggr)+\partial _{y}g \frac{\partial _{y}\theta }{g} \\ ={}&\frac{g}{\kappa } \biggl(c_{v}\rho _{0}\partial _{t}\theta - \partial _{y}uF - \frac{1}{2} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2}\partial _{y}u - g \biggl\vert \frac{1}{g} \biggl(\frac{d_{y}}{g} \biggr)_{y} + \frac{1}{g^{2}} \vert d_{y} \vert ^{2}d\biggr\vert ^{2} \biggr) \\ &{}+\partial _{y}g\frac{\partial _{y}\theta }{g}, \end{aligned}$$
and thus, by the Hölder, Young, and Gagliardo–Nirenberg inequalities and (2.88), it follows from Corollaries 3–1 and Proposition 7 that
$$\begin{aligned} \bigl\Vert \partial _{y}^{2}\theta \bigr\Vert _{2} \leq & C\biggl( \Vert \sqrt{\rho _{0}}\partial _{t} \theta \Vert _{2}+ \Vert \partial _{y}u \Vert _{2} \biggl( \Vert F \Vert _{\infty }+ \biggl\Vert \bigg\vert \frac{d_{y}}{g} \bigg\vert ^{2} \biggr\Vert _{\infty } \biggr) + \Vert g \Vert _{\infty } \Vert d_{t} \Vert _{2}^{2} \\ &{}+ \Vert \partial _{y}g \Vert _{2} \Vert \partial _{y}\theta \Vert _{\infty }\biggr) \\ \leq & C \biggl(1+ \Vert g_{0} \Vert _{2}+ \Vert h_{0} \Vert _{2}+ \Vert F \Vert _{H^{1}}+ \biggl\Vert \bigg\vert \frac{d_{y}}{g} \bigg\vert ^{2} \biggr\Vert _{\infty }+ \Vert \partial _{y}\theta \Vert _{2}^{\frac{1}{2}} \bigl\Vert \partial _{y}^{2} \theta \bigr\Vert _{2}^{ \frac{1}{2}} \biggr) \\ \leq &\frac{1}{2} \bigl\Vert \partial _{y}^{2} \theta \bigr\Vert _{2}+C\bigl(1+ \Vert h_{1} \Vert _{2}+ \Vert h_{2} \Vert _{2}\bigr), \end{aligned}$$
(2.94)
which gives
$$ \sup_{0\leq t\leq T} \bigl\Vert \partial _{y}^{2} \theta \bigr\Vert _{2}^{2}\leq C\bigl(1+ \Vert h_{1} \Vert _{2}^{2}+ \Vert h_{2} \Vert _{2}^{2}\bigr) $$
(2.95)
for a positive constant C depending only on R, \(c_{v}\), μ, κ, \(m_{1}\), \(\mathcal{N}_{1}\), \(\mathcal{N}_{2}\), and T.
By calculations, one deduces
$$\begin{aligned} \partial _{y}^{2} p =&R\partial _{y}^{2} \biggl(\frac{\rho _{0}}{g} \theta \biggr) \\ =&R \biggl[\rho _{0}'' \frac{\theta }{g}+2\rho _{0}' \partial _{y} \biggl(\frac{\theta }{g} \biggr)+\rho _{0}\partial _{y}^{2} \biggl(\frac{\theta }{g} \biggr) \biggr] \\ =&R \biggl[\rho _{0}''\frac{\theta }{g} +2\rho _{0}' \biggl( \frac{\partial _{y} \theta }{g}- \frac{\partial _{y}g}{g^{2}}\theta \biggr) \\ &{}+\rho _{0} \biggl(\frac{\partial _{y}^{2}\theta }{g}- \frac{2}{g^{2}} \partial _{y} g\partial _{y}\theta +2 \frac{(\partial _{y}g)^{2}}{g^{3}} \theta - \frac{\partial _{y}^{2}g}{g^{2}}\theta \biggr) \biggr]. \end{aligned}$$
Therefore, by the Hölder and Sobolev embedding inequalities, using (2.88), (2.89), and (2.95), it follows from Corollary 3 and Corollary 1 that
$$\begin{aligned} \bigl\Vert \partial _{y}^{2} p \bigr\Vert _{2} \leq &C \bigl[ \bigl\Vert \rho _{0}'' \bigr\Vert _{2} \Vert \theta \Vert _{\infty }+2 \bigl\Vert \rho _{0}' \bigr\Vert _{\infty } \bigl( \Vert \partial _{y}\theta \Vert _{2} + \Vert \partial _{y} g \Vert _{2} \Vert \theta \Vert _{\infty } \bigr) \\ &{}+ \Vert \rho _{0} \Vert _{\infty } \bigl( \bigl\Vert \partial _{y}^{2}\theta \bigr\Vert _{2}+2 \Vert \partial _{y} g \Vert _{\infty } \Vert \partial _{y} \theta \Vert _{2}+2 \Vert \partial _{y}g \Vert _{\infty } \Vert \partial _{y}g \Vert _{2} \Vert \theta \Vert _{\infty } \bigr) \\ &{} + \Vert \rho _{0} \Vert _{\infty } \bigl\Vert \partial _{y}^{2}g \bigr\Vert _{2} \Vert \theta \Vert _{\infty } \bigr] \\ \leq & C\bigl(1+ \Vert h_{1} \Vert _{2}+ \Vert h_{2} \Vert _{2} + \bigl\Vert \partial _{y}^{2}g \bigr\Vert _{2}\bigr) \end{aligned}$$
(2.96)
for a positive constant C depending only on R, \(c_{v}\), μ, κ, \(m_{1}\), \(\mathcal{N}_{1}\), \(\mathcal{N}_{2}\), \(\mathcal{N}_{3}\), and T.
Using (2.86) and (1.8)1, we deduce
$$\begin{aligned} \partial _{y}^{3}\theta ={}& \partial _{y}^{2} \biggl( \frac{\partial _{y}\theta }{g} \biggr) g +2\partial _{y} \biggl( \frac{\partial _{y}\theta }{g} \biggr)\partial _{y}g + \frac{\partial _{y}\theta }{g}\partial _{y}^{2}g \\ ={}&\frac{g}{\kappa } \biggl[c_{v} \bigl(\rho _{0}\partial _{t}\partial _{y} \theta +\rho _{0}' \partial _{t}\theta \bigr) -\partial _{y}u \partial _{y}F-\partial _{y}^{2}uF -\frac{1}{2} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2}\partial _{y}^{2}u -\partial _{y}u \frac{d_{y}}{g}\cdot \biggl(\frac{d_{y}}{g} \biggr)_{y} \\ &{}-2gd_{t}\cdot d_{ty}- \vert d_{t} \vert ^{2}g_{y} \biggr] \\ &{}+\frac{2}{\kappa }\partial _{y}g \biggl(c_{v}\rho _{0}\partial _{t} \theta - \partial _{y}uF - \frac{1}{2} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \partial _{y}u - g \vert d_{t} \vert ^{2} \biggr)+ \frac{\partial _{y}\theta }{g}\partial _{y}^{2}g. \end{aligned}$$
Therefore, by the Hölder and Sobolev embedding inequalities, using (2.88), (2.90), (2.93), (2.95), (2.96), Corollary 3, Corollary 1, and (ii) of Propositions 6, 7, it follows
$$\begin{aligned} \bigl\Vert \partial _{y}^{3}\theta \bigr\Vert _{2}\leq{} &C \biggl[ \Vert \partial _{y} \partial _{t} \theta \Vert _{2} + \bigl\Vert \rho _{0}' \bigr\Vert _{2} \Vert \partial _{t}\theta \Vert _{\infty }+ \Vert \partial _{y}u \Vert _{\infty } \biggl( \Vert \partial _{y} F \Vert _{2}+ \biggl\Vert \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{2} \biggr) \\ &{}+ \bigl\Vert \partial _{y}^{2}u \bigr\Vert _{2} \biggl( \Vert F \Vert _{\infty }+ \biggl\Vert \bigg\vert \frac{d_{y}}{g} \bigg\vert ^{2} \biggr\Vert _{\infty } \biggr)+ \Vert d_{t} \Vert _{\infty } \Vert d_{ty} \Vert _{2}+ \Vert d_{t} \Vert _{2}^{2} \Vert g_{y} \Vert _{\infty } \\ &{}+ \Vert \partial _{y}g \Vert _{\infty } \biggl( \Vert \sqrt{ \rho _{0}}\partial _{t} \theta \Vert _{2}+ \Vert \partial _{y}u \Vert _{2} \biggl( \Vert F \Vert _{\infty }+ \biggl\Vert \bigg\vert \frac{d_{y}}{g} \bigg\vert ^{2} \biggr\Vert _{\infty } \biggr) + \Vert d_{t} \Vert _{2}^{2} \biggr) \\ &{}+ \Vert \partial _{y}\theta \Vert _{\infty } \bigl\Vert \partial _{y}^{2}g \bigr\Vert _{2} \biggr] \\ \leq{}& C \biggl[ \Vert \partial _{y}\partial _{t} \theta \Vert _{2}+ \Vert \sqrt{\rho _{0}} \partial _{t}\theta \Vert _{2}+ \Vert \partial _{y}u \Vert _{H^{1}} \biggl( \Vert \partial _{y} F \Vert _{2}+ \biggl\Vert \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{2} \biggr) \\ &{}+ \bigl\Vert \partial _{y}^{2}u \bigr\Vert _{2} \biggl( \Vert F \Vert _{H^{1}}+ \biggl\Vert \bigg\vert \frac{d_{y}}{g} \bigg\vert ^{2} \biggr\Vert _{\infty } \biggr)+ + \Vert \partial _{y}\theta \Vert _{H^{1}} \bigl\Vert \partial _{y}^{2}g \bigr\Vert _{2} + \Vert d_{ty} \Vert _{2} \\ &{}+ \Vert \partial _{y}g \Vert _{H^{1}} \biggl( \Vert \sqrt{ \rho _{0}}\partial _{t} \theta \Vert _{2}+ \Vert \partial _{y}u \Vert _{2} \biggl( \Vert F \Vert _{H^{1}} + \biggl\Vert \bigg\vert \frac{d_{y}}{g} \bigg\vert ^{2} \biggr\Vert _{\infty } \biggr) + \Vert d_{t} \Vert _{2}^{2} \biggr) \biggr] \\ \leq{}& C\bigl(1+ \Vert h_{1} \Vert _{2}+ \Vert h_{2} \Vert _{2}+ \Vert \partial _{y}\partial _{t} \theta \Vert _{2}+ \bigl\Vert \partial _{y}^{2}g \bigr\Vert _{2}\bigr) \\ \leq{}& C\bigl(1+ \Vert h_{1} \Vert _{2}+ \Vert h_{2} \Vert _{2}\bigr) \bigl(1+ \Vert \partial _{y}\partial _{t} \theta \Vert _{2}+ \bigl\Vert \partial _{y}^{2}g \bigr\Vert _{2}\bigr) \end{aligned}$$
(2.97)
and
$$\begin{aligned} \partial _{y}^{3}u =& \partial _{y}^{2} \biggl( \frac{\partial _{y}u}{g} \biggr) g +2\partial _{y} \biggl( \frac{\partial _{y} u}{g} \biggr)\partial _{y}g + \frac{\partial _{y} u}{g}\partial _{y}^{2}g \\ =&\frac{g}{\mu } \biggl(\rho _{0}'\partial _{t} u+\rho _{0}\partial _{t} \partial _{y} u +\partial _{y}^{2} p + \frac{d_{y}}{g} \cdot \biggl( \frac{d_{y}}{g} \biggr)_{y} \biggr) \\ &{}+\frac{2}{\mu }\partial _{y} g \biggl(\rho _{0} \partial _{t} u + \partial _{y} p+ \frac{1}{2} \biggl\vert \frac{d_{y}}{g} \biggr\vert ^{2} \biggr)+ \frac{\partial _{y} u}{g}\partial _{y}^{2}g, \end{aligned}$$
which is estimated as
$$\begin{aligned} \bigl\Vert \partial _{y}^{3}u \bigr\Vert _{2} \leq & C \biggl[ \biggl( \Vert \partial _{t}u \Vert _{\infty }+ \Vert \partial _{y}\partial _{t}u \Vert _{2} + \bigl\Vert \partial _{y}^{2} p \bigr\Vert _{2} + \biggl\Vert \biggl(\frac{d_{y}}{g} \biggr)_{y} \biggr\Vert _{2} \biggr) \\ &{}+ \Vert \partial _{y}g \Vert _{\infty } \biggl( \Vert \sqrt{\rho _{0}}\partial _{t}u \Vert _{2}+ \Vert \partial _{y} p \Vert _{2}+ \biggl\Vert \frac{ \vert d_{y} \vert }{\sqrt{g}} \biggr\Vert _{2}^{2} \biggr) + \Vert \partial _{y} u \Vert _{\infty } \bigl\Vert \partial _{y}^{2} g \bigr\Vert _{2} \biggr] \\ \leq &C \bigl[ \Vert \partial _{y}\partial _{t}u \Vert _{2}+ \Vert \sqrt{\rho _{0}} \partial _{t}u \Vert _{2}+ 1+ \Vert h_{1} \Vert _{2}+ \Vert h_{2} \Vert _{2}+ \bigl\Vert \partial _{y}^{2}g \bigr\Vert _{2} \\ &{}+ \Vert \partial _{y}g \Vert _{H^{1}}\bigl(1+ \Vert h_{1} \Vert _{2}+ \Vert h_{2} \Vert _{2}\bigr)+ \Vert \partial _{y}u \Vert _{H^{1}} \bigl\Vert \partial _{y}^{2} g \bigr\Vert _{2} \bigr] \\ \leq &C\bigl(1+ \Vert h_{1} \Vert _{2}+ \Vert h_{2} \Vert _{2}\bigr) \bigl(1+ \Vert \partial _{y}\partial _{t}u \Vert _{2}+ \bigl\Vert \partial _{y}^{2}g \bigr\Vert _{2}\bigr) \end{aligned}$$
(2.98)
for a positive constant C depending only on R, \(c_{v}\), μ, κ, \(m_{1}\), \(\mathcal{N}_{1}\), \(\mathcal{N}_{2}\), \(\mathcal{N}_{3}\), and T.
Combining (2.97) with (2.98) and using (2.91), one obtains
$$\begin{aligned} \int _{0}^{t} \bigl\Vert \bigl(\partial _{y}^{3}u,\partial _{y}^{3}\theta \bigr) \bigr\Vert _{2}^{2}\,d\tau \leq & C\bigl(1+ \Vert h_{1} \Vert _{2}^{2}+ \Vert h_{2} \Vert _{2}^{2}\bigr) \int _{0}^{t}\bigl(1+ \Vert \partial _{y}\partial _{t} u \Vert _{2}^{2}+ \bigl\Vert \partial _{y}^{2}g \bigr\Vert _{2}^{2}\bigr)\,d\tau \\ \leq & C\bigl(1+ \Vert h_{1} \Vert _{2}^{2}+ \Vert h_{2} \Vert _{2}^{2}\bigr)^{2} \biggl(1+ \int _{0}^{t} \bigl\Vert \partial _{y}^{2}g \bigr\Vert _{2}^{2}\,d\tau \biggr) \end{aligned}$$
(2.99)
for any \(t\in [0,T]\), where C is a positive constant depending only on R, \(c_{v}\), μ, κ, \(m_{1}\), \(\mathcal{N}_{1}\), \(\mathcal{N}_{2}\), \(\mathcal{N}_{3}\), and T. Using (1.8)1, one gets \(g=1+\int _{0}^{t}\partial _{y}u\,d\tau \), and thus it follows from the Hölder inequality that
$$\begin{aligned} \bigl\Vert \partial _{y}^{2}g \bigr\Vert _{2}^{2}(t) =& \biggl( \biggl\Vert \int _{0}^{t} \partial _{y}^{3}ud \tau \biggr\Vert _{2} \biggr)^{2}\leq \biggl( \int _{0}^{t} \bigl\Vert \partial _{y}^{3}u \bigr\Vert _{2}\,d\tau \biggr)^{2} \leq t \int _{0}^{t} \bigl\Vert \partial _{y}^{3}u \bigr\Vert _{2}^{2}\,d\tau . \end{aligned}$$
Combining this with (2.99) and applying the Gronwall inequality, one obtains
$$\begin{aligned} \int _{0}^{T}\bigl( \bigl\Vert \partial _{y}^{3}u \bigr\Vert _{2}^{2}+ \bigl\Vert \partial _{y}^{3} \theta \bigr\Vert _{2}^{2}\bigr)\,dt\leq C, \end{aligned}$$
and, further, that
$$\begin{aligned} \sup_{0\leq t\leq C} \bigl\Vert \partial _{y}^{2}g \bigr\Vert _{2}^{2}\leq C \end{aligned}$$
for a positive constant C depending only on R, \(c_{v}\), μ, κ, \(m_{1}\), \(\mathcal{N}_{1}\), \(\mathcal{N}_{2}\), \(\mathcal{N}_{3}\), and T. □