Global existence of classical solutions for two-dimensional isentropic compressible Navier–Stokes equations with small initial mass

In this paper, we consider the initial-boundary value problem of two-dimensional isentropic compressible Navier–Stokes equations with vacuum on the square domain. Based on the time-weighted uniform estimates, we prove that the classical solution exists globally in time if the initial mass ∥ρ0∥L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\|\rho_{0}\|_{L^{1}}$\end{document} of the fluid is small. Here, we do not require the initial energy or the upper bound of the initial density to be small.


Introduction
In this paper, we consider the following two-dimensional isentropic compressible Navier-Stokes equations in the Eulerian coordinates: ⎧ ⎨ ⎩ ρ t + div(ρu) = 0, (ρu) t + div(ρu ⊗ u)μ u -(μ + λ)∇ div u + ∇P(ρ) = 0, (1) where t ≥ 0 is the time, x ∈ Ω = [0, 1] × [0, 1] is a spatial coordinate. ρ = ρ(x, t), u = (u 1 , u 2 )(x, t) and case, Nash [1] and Itaya [2] established the local existence and uniqueness of classical solutions in the absence of vacuum in 1962 and 1977, respectively. In 1995, Hoff [3,4] proved the global existence of weak solutions when the initial density would be close to a constant in L 2 and L ∞ norm, and the initial velocity be small in L 2 norm and bounded in L 2 n norm (n is the space dimension). In 1998, Lions [5] obtained the global existence of weak solutions when the adiabatic exponent γ is suitably large, the main restriction on initial data is that the initial total energy is finite, similar results can be found in [6] given by Feireisl. A few years later, Hoff [7][8][9] obtained a new type of global weak solutions with small energy, which have more regularity information than the works in [5,6]. On the other hand, when vacuum is allowed, Cho and Kim [10,11] proved the existence of unique local strong solutions in bounded and unbounded domains in 2003. In 2012, Huang, Li and Xin [12] established the global classical solutions with small energy but possibly large oscillations. In the same year, Duan [13] generalized the result in [7] and proved the global classical solutions to the half-space problem with the boundary condition proposed by Navier provided the initial energy is small. In 2016, Yu and Zhang [14] studied the nonhomogeneous equations with density-dependent viscosity in a smooth bounded domain and the vacuum is allowed. The global well-posedness of strong solutions is established for the case when the bound of the density is suitably small, or when the total mass is small with large oscillations. Later, in 2017, under the same condition in [12], Yu and Zhao [15] studied the global existence in a cuboid domain, some new ideas being applied to establishing a time-uniform upper bound for the density. Recently, Si, Zhang and Zhao [16] established the global existence of classical solutions with a small initial density but possibly a large energy in the case of ρ 0 ∈ L γ , γ ∈ (1, 6) and ρ 0 ∈ L 1 , γ > 1, respectively, which extends the results in [12].
Compared with the three-dimensional case, there are few results in the two-dimensional space. The pioneering work can be traced back to [17] in 1995, as Vaigant and Kazhikhov first proposed the initial-boundary value problem with the special viscosity coefficients, that is, shear viscosity μ being a positive constant and bulk viscosity λ(ρ) = ρ β , β > 3. (2) They proved the existence of global strong solution with no restrictions on the size of initial data. In 2012, Luo [18] studied the Cauchy problem and proved local existence and uniqueness of classical solutions with initial density containing vacuum when viscosity coefficients μ and λ are constant. For the case of a viscosity depending on the density, we refer to a later work by Li and Liang [19]. In 2013, under the condition (2), Jiu, Wang and Xin [20,21] proved the global classical solutions on the torus and in the whole space, respectively, where the initial data may contain vacuum in an open set. In the same year, Ducomet and Necasova [22] studied the initial-boundary value problem with a vorticitytype boundary condition and prove that the results of [17] hold in any smooth bounded domain. In 2014, Zhang, Deng and Zhao [23] established the global classical solutions to the Cauchy problem with smooth initial data under the assumption that the viscosity coefficient μ is large enough. In 2016, Huang and Li [24] relaxed the power index β in (2) to be β > 4 3 and studied the large-time behavior of the solutions, also see a recent work [25] for Cauchy problem. In the same year, Fang and Guo [26] established the global existence and large-time asymptotic behavior of the strong solution to the Cauchy problem in the case of β ∈ [0, 1] provided that the initial data are of small total energy. In 2018, Ding, Huang and Liu [27] obtained the global classical solutions to the Cauchy problem with β ∈ [0, 1] under the condition of small initial density, which extends the earlier work [26] with small initial energy.
From the well-known results mentioned in the above paragraph, we can see that in the two-dimensional space, the existing work mainly discussed the global existence of system (1) under the condition of density-dependent viscosity, β > 4 3 with general initial data, β ∈ [0, 1] with small initial energy or small initial density. However, whether the unique local classical solution can exist globally for constant viscosity with small initial mass on a bounded domain is still unknown at present. Inspired by the analysis of [12] and [15], in this paper, we consider Dirichlet problem of (1) with the following initial-boundary conditions: We hope to establish the global existence of strong solutions for (1), (3)-(4) with constant viscosity on the square domain.
Before stating the main results, we explain the notations and conventions used throughout this paper. Notations: • The standard Lebesgue and Sobolev spaces are defined as follows: •ḟ = f t + u · ∇f denotes the material derivative of f .
• The symbol ∇ l with an integer l ≥ 0 stands for the usual spatial derivatives of any order l. We define • Positive generic constants are denoted by C, which may change in different places. Now, our main results in this paper can be stated as follows.
Theorem 1.1 For given numbersρ > 0, M > 0 and q > 2, suppose that the initial data and the following compatibility conditions: for some g ∈ L 2 . Then, there exists a positive constant ε 0 depending onρ, M, μ, λ, and some other known constants but independent of T , such that, if the initial-boundary value problem (1), (3)-(4) admits a unique global classical solution (ρ, u) in Ω × (0, +∞) satisfying, for any 0 < T < +∞, Remark 1.1 Cho and Kim [10,11] proved the existence and uniqueness of local strong solution to (1), (3)-(4) with initial vacuum in the three-dimensional space, where Ω can be bounded domain or the whole space. If Ω is a bounded domain in R 2 and the initial data (ρ 0 , u 0 ) are smooth enough, and u satisfies the boundary condition (4), it is not difficult to verify that the proofs in [10,11] are still valid for local existence of classical solutions in two-dimensional space.
Remark 1.2 In Theorem 1.1, we give the global existence of classical solution to the initialboundary value problem (1), (3)-(4) provided the initial mass ρ 0 L 1 is small. In fact, if we take the same vorticity-type boundary condition (Navier-slip boundary condition) in [15] instead of Dirichlet boundary condition, by applying the same method in threedimensional space, similar results in Theorem 1.1 can also be proved. Thus, our results extended the one due to Yu and Zhao [15], where the global well-posedness of classical solutions with small initial energy was proved. Moreover, under the condition (7), we can prove the global existence of classical solution to the Cauchy problem in threedimensional space by using effective viscous flux method, which extend the results of [12] for small initial energy and [16] for small initial density.
We now make some comments on the global existence of classical solution to the isentropic compressible Navier-Stokes equations. Compared with the three-dimensional case, it causes some essential difficulties. Similar to the procedure of [12,15,16], a key ingredient in our proof is to obtain a uniform priori upper bound for the density function. However, due to the invalidity of the Sobelov embedding inequality u L 6 ≤ C ∇u L 2 , and there is no boundary information of effective viscous flux F (2μ + λ) div u -P in the two-dimensional bounded domain, time-weighted estimates are needed the ensure the better integrability of the velocity, which is quite differs from three-dimensional Cauchy problem. In this paper, we use the Poincaré inequality and the following decomposition of the velocity u = v + w to overcome this difficulty, where v solves the elliptic system: Then, from the momentum, Eqs. (1) and (9), we can see that w satisfies Hence, ∇u L p , p ≥ 2, is controlled by the standard L p -estimate of elliptic system (9) and (10).
On the one hand, under the condition of (7), we have the following key observation: which is derived from (1) 1 and (1) 2 . Then, by applying the method in [15], we get the uniform bound for ∇u L 2 and time-dependent bound for ∇ u L 2 (t 1 ,t 2 ;L 2 ) , by which, together with Zlotnik inequality, we have the uniform upper bound of density. It is worth mentioning that, these boundness can be obtained by the smallness of the initial mass ρ 0 L 1 instead of the smallness of the upper bound of the density in [16] and the initial energy in [12,15], respectively. At last, higher-order regularity estimates for (ρ, u) can be proved by standard methods after some modifications, see [12] for example. Finally, after all the required a priori estimates obtained, by using the continuity argument, we can extend the local classical solution to a global one. The rest of the paper is organized as follows: In Sect. 2, we list some elementary inequalities which will be used in later analysis. Section 3 is devoted to deriving the necessary a priori estimates on classical solution which extend the local solution to a global one.

Preliminaries
In this section, we recall some well-known inequalities, which will be used frequently throughout this paper. First, we give the Sobolev-Poincaré lemma [28].

Lemma 2.1 There exists a positive constant C depending only on Ω such that every func-
Next, we give some regularity results for the following Lamé system with the Dirichlet boundary condition (see [29]): Suppose U ∈ H 1 0 is a weak solution to the Lamé system, we could denote U = L -1 F due to the uniqueness of solution.

Lemma 2.2
Let r ∈ (1, +∞), then there exists some generic constant C > 0 depending only on μ, λ, r and Ω such that ( where BMO(Ω) stands for the John-Nirenberg space of mean oscillation whose norm is defined by In the following, we give two critical Sobolev inequalities of logarithmic type, which are originally due to Brezis-Gallouet [30] and Brezis-Wainger [31].

Lemma 2.3
Let Ω ∈ R 2 be a bounded Lipschitz domain and f ∈ W 1,q with q > 2, then we have with a constant C depending only on q.

Lemma 2.4
Let Ω ∈ R 2 be a smooth domain and f ∈ L 2 (s, t; H 1 0 ) ∩ L 2 (s, t; W 1,q ), with some q > 2 and 0 ≤ s < t ≤ ∞. Then we have with a constant C depending only on q.
Finally, we give the following lemma arises from Zlotnik [32], which will be used to prove the uniform upper bound for the density.

Global classical solution
In this section, we establish some necessary a priori estimates for the classical solutions of initial-boundary value problem (1), (3)-(4). We assume that, for any T > 0, let (ρ, u) be a classical solution of (1), (3)-(4) in the solution space (8) with the initial data satisfying (5) and (6). In Sects. 3.1 and 3.2, we will show the lower-order and the higher-order estimates of the solutions, which guarantee the local classical solution can be extended to a global one.

Lower-order estimates of the solutions
First, we give the following proposition to prove the uniform upper bounds of ∇u L 2 and ρ.
Proposition 3.1 Assume that the initial data satisfy (5)- (6), and the local classical solution satisfies where Then there exists depending onρ, M, μ, λ, and some other known constants but independent of T such that sup 0≤t≤σ (T) provided that m 0 ≤ ε 2 is suitable small.
In order to prove Proposition 3.1, we give the following mass conservation identity and the uniform bound of ∇u L 2 (0,T;L 2 ) , which are the foundation of our proof in this paper.
provided there exists a positive constant ε 1 such that m 0 ≤ ε 1 .
Next, in Lemma 3.3, we give the uniform upper bound of ∇u L 2 .
In Lemma 3.4, we will give the bound for t 2 t 1 σ 2 ∇u 2 L 2 ds which will be used to prove the uniform upper bound of ρ. It should be noted that the constant C on the right-hand side of (39) and (40) is independent of time.
for any t 1 , Proof Operating ηu j (∂ t + div(u·)) to (1) j 2 , summing with respect to j, and integrating the resulting equation over Ω, we obtain It follows from integration by parts and using Eq. (1) 1 that Similarly, we get where ∇u 4 L 4 can be estimated as Substituting the estimates J 1 , J 2 , J 3 and (45) into (41), we arrive at In order to prove (39), taking η = σ 2 i in (46), integrating (46) over (i -1, t) and taking (27) into consideration, we get which proves (39). Furthermore, from (47), we can see that, if we take η = σ 2 , then integrating (46) over (t 1 , t 2 ) ∈ [0, T], we have ≤ Cm where we have used (23), (20) and (39). In order to estimate the second term on the right-hand side of the above inequality, taking η = σ in (29), integrating (29) over (t 1 , t 2 ) ∈ [0, T], we get Inspired by the methods in Refs. [12,15], in the following lemma, we use the Zlotnik inequality to prove the uniform upper bound of the density ρ.
Proof For any given (x, t) ∈ Ω × [0, T], denoting X(s; x, t) the solution to the initial value problem d ds X(s; x, t) = u(X(s; x, t), s), 0 ≤ s < t, It is easy to verify that d ds ρ X(s; x, t), s + ρ X(s; x, t), s div u X(s; x, t), s = 0, due to (1) 1 . This gives where and C(t) = (2μ + λ)divv -P. Next, we use Lemma 2.5 to prove the uniform upper bound of the density. We have In the following, we estimate the terms on the right-hand side of Eq. (53) one by one. In order to estimate C(t), from Eq. (9), we have ) and the boundary condition (4) implies Then, we have (∇ × (∇ × v)) · n = 0 a.e. on ∂Ω and div(∇ × (∇ × v)) = 0. Multiplying (54) by ∇((2μ + λ) div v -P) and integrating the resulting equation over Ω, we arrive at ∇ (2μ + λ) div v -P L 2 = 0, which implies that there exists C(t) such that Using (9), we have ∇v L 2 ≤ C P L 2 . Integrating (56), we get Then, we have provided there exists a constant ε * 3 , m 0 ≤ ε * 3 such that In order to estimate K 2 , we consider the following three cases: where we have used Lemma 3.3. It remains to estimate the term on the right-hand side of inequality (59). To do this, we taking η = σ in (46) and integrating the resulting inequality over (0, σ (T)), we have From (59) and (60), we can see that provided there exists a constant ε * 4 such that m 0 ≤ ε * 4 .
Proof Taking η = 1 in (46), integrating (46) over (0, σ (T)], we get which combines (39) and (40), and we obtain Next, applying the operator ∇ to (1) 1 , and multiplying the resulting equation by p|∇ρ| p-2 ∇ρ, p > 2, we obtain then we have where the terms on the right-hand side of the above inequality can be estimated as and Inserting (72) and (73) into (71), we have Both sides of (74) divided by ∇ρ p + e lead to d dt ln ∇ρ L p + e ≤ C ∇u L 2 + e ln e + ∇ρ L p + C ∇u L 2 + e .
Then, by using the Gronwall inequality and (69), we have Moreover, from (69), we have This completes the proof of Lemma 3.6.

Higher-order estimates of the solutions
For completeness of our proof, we list the higher-order estimates of the solution (ρ, u) below, which can be derived in a similar manner to those obtained in [12] after some modifications.
Proof Estimate (78) follows directly from the following simple facts that where in the last inequality we have used the Sobolev embedding inequalities and Lemma 3.6. Next, we prove (79). P satisfies which together with (1) 1 yields where in the last inequality we have used (72) and the following standard L 2 -estimate for elliptic system (9) and (10) Then, combining (83), Lemma 3.6 and the Gronwall inequality, we have (79).
Proof First, from (82) and Lemma 3.6, we obtain Furthermore, differentiating (82) yields which together with Lemma 3.6 and Lemma 3.7, one gets The combination of (86) and (88) implies Note that P tt satisfies P tt + u t · ∇P + u · ∇P t + γ P div u t + γ P t div u = 0, from which, together with (89) and Lemma 3.7, we have Next, we differentiate (1) 2 with respect to t, then multiplying the resulting equation by u tt , one gets after integration by parts The terms on the right-hand side of Eq. (92) can be estimated as follows: where we have used Lemma 3.6, (84) and the Poincaré inequality. We have At last, integrating (92) over time (0, T), and inserting estimates (93)-(97), we have from which, together with the Gronwall inequality, one obtains (85) immediately. This completes the proof of Lemma 3.8.

Lemma 3.9
Let (ρ, u) be a classical solution of (1), (3)-(4) on Ω × (0, T], under the condition of Theorem 1.1, the following estimates hold: Proof It follows from Lemma 3.8 that which together with Lemma 3.6 gives The standard H 1 estimate for elliptic system (28) yields Then, as a consequence of (67) and (103), we have Moreover, the standard L 2 -estimate for elliptic system (28) and Lemma 3.8 yield which together with (85) implies On the other hand, applying the standard H 2 -estimate for the elliptic system (28) again leads to where and In order to estimate the third term on the right-hand side of (107), applying ∇ 3 to (82) and integrating the resulting equation over Ω, we obtain which, together with the Gronwall inequality (106), implies that sup 0≤t≤T ∇ 3 P L 2 ≤ C.
Taking (106)-(111) into consideration, we have It is easy to check that similar arguments work for ρ by using (112). Hence the proof of Lemma 3.9 is completed.
Finally, by using the continuity argument, we can extend the local classical solution to a global one, and thus Theorem 1.1 is proved.