Persistence of the incompressible Euler equations in a Besov space
© Pak and Park; licensee Springer 2013
Received: 8 April 2012
Accepted: 10 May 2013
Published: 30 May 2013
The unique existence of a solution of the incompressible Euler equations in a critical Besov space for is investigated. The global existence of a solution of two-dimensional Euler equations is also discussed.
1 Main theorems and terminology
are considered. Here is the Eulerian velocity of a fluid flow and , with .
The best local existence and uniqueness results known for the Euler equations (1) in Besov spaces are a series of theorems on the space with (see the introductions in [1, 2] for details and the references therein). The local existence for the limit case of has not been reported yet possibly due to the lack of -estimates. On the other hand, the ill-posedness of the Euler equations in  for a range of Besov spaces has been recently studied, which signifies that it is worthwhile to clarify either the well-posedness or the ill-posedness of the solutions in some particular Besov spaces. This is why the existence problem in the space is not trivial even though it is smaller than the space .
This paper takes care of the local unique existence of the solution to the Euler equations (1) in a critical Besov space and of the global existence for a two-dimensional case. Our main results are the following.
Theorem 1.1 (Local existence and uniqueness)
For any divergence-free vector field , there exists a positive time T for which the initial value problem (1) with has a unique solution u in the space .
Theorem 1.2 (2-D global existence)
For any divergence-free vector field , there exists a unique solution of the problem (1) with .
The compactness argument was used for the main literature in the proof employed by the authors in , and one of the primary difficulties of estimates was to get rid of the -singularity. To take care of it, we present some new estimates together with substantial modifications of the identities and of the estimates proved in . The essential tools for a priori estimates are Bony’s para-product formula and Littlewood-Paley decomposition. An Osgood-type ordinary differential inequality is solved to complete the compactness argument.
For the proof of global existence in two-dimensional case, the limiting case of Beale-Kato-Majda inequality in 2-D is fetched, which is a modification of the inequality originally proved in  by Vishik. In , Chae proved the global existence of velocity in the Triebel-Lizorkin spaces , , for the 2-D Euler equations, and also discussed the vorticity existence in the spaces , or ∞. Since the spaces are equivalent to the spaces , our 2-D global velocity existence theorem is similar to Chae’s theorem displayed in .
The corresponding spaces of vector-valued functions are denoted by the bold faced symbols. For example, the product space is denoted by and the corresponding triple Besov spaces by . Note that the classical Hölder spaces are equivalent to the Besov spaces (if ); see, for example, p.26 in  or .
Notation Throughout this paper (especially in Section 4), the notation means that , where C is a fixed but unspecified constant. Unless explicitly stated otherwise, C may depend on the dimension d and various other parameters (such as exponents), but not on the functions or variables involved.
2 Local existence and uniqueness of the solution
Then we first note that for any . Therefore, for fixed , classical results (see ) say that for each m, there exist a maximal time and a solution to the Euler equations (2) in . In case of dimension 2, it is well-known that . That is, there is a global solution .
2.1 Compactness of the sequence
and from the fact that , we see that is a lower bound of .
We close this section by explaining the continuity of on with values in the Besov space .
Lemma 2.1 (Temporal regularity)
Suppose that v is a solution for the Euler equations (1) staying inside of with initial velocity . Then v is continuous on with values in , that is, .
Proof First, applying Propositions 4.2 and 4.5 in Section 4 to the Euler equations (1), we can deduce that , and so .
The first term of the right-hand side converges to zero as ℓ tends to infinity because . By virtue of Propositions 4.4 and 4.5 in Section 4 and the fact that , the second and third terms of the right-hand side also converge to zero as ℓ tends to infinity. Hence the sequence converges to v in .
together with the fact that , we can deduce that each is continuous on with values in . In all, the limit v is continuous on with values in . □
From this lemma, we observe that . We also notice that the sequence is uniformly bounded in , thanks to Propositions 4.2 and 4.5.
2.2 Convergence of the sequence
This implies that is a Cauchy sequence in . Hence there exists a strong limit u of the sequence in the space .
2.3 Local existence of a solution
Due to the fact that the two limits should coincide, we have , and . This implies that u is absolutely continuous on with values in .
So far, we have shown that u is a solution for the Euler equations (1) located in . Hence, by virtue of Lemma 2.1, we conclude that u belongs to . We may continue to use this argument until the value blows up, that is, . This completes the proof of the local existence.
2.4 Uniqueness of a solution
which in turn implies the uniqueness of a solution for (1) in .
3 2-D global existence - Proof of Theorem 1.2
The estimate (15) suggests that we focus on proving that does not blow up for all time. For this, we recall the limiting case of Beale-Kato-Majda inequality in .
Proposition 3.1 (Vishik’s inequality)
The original version of the proposition was proved by Vishik in the space in , and Chae later generalized it to the Besov spaces and the Triebel-Lizorkin spaces in . Our version (in ) can be considered as a slight generalization of those, and the proof is almost the same as the original proof except for inserting the differential index .
where the constant depends only on (and so ). This completes the 2-D global existence of a solution in .
4 A priori estimates
where represents Bony’s para-product of f and g defined by and denotes the remainder of the para-product . The estimates of para-product parts in are provided as follows.
Lemma 4.1 (Para-product estimate)
The para-product estimate implies the following pointwise product estimate in .
Proposition 4.2 (Product formula)
This also implies the second inequality in the statement. □
Here is a commutator estimate in .
Lemma 4.3 (Commutator estimate)
where the commutator is defined as .
We now present the primary estimates which have been used for the proof of the main theorem.
The estimates (16) and (17) for are obtained simply by using the commutator estimate (Lemma 4.3). By putting estimates together, the desired results can be achieved. □
and the estimate (29) yield the first estimate (26). Also, the second estimate (27) follows from the estimate (30) together with (33). The last estimate (28) follows from the symmetry of π; . □
The first author was supported by the research fund of Dankook University in 2011.
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