Global well-posedness of 3-D anisotropic Navier-Stokes system with large vertical viscous coefficient
Introduction
In this paper, we first investigate the global well-posedness of the following 3-D anisotropic Navier-Stokes system provided that the vertical viscous coefficient is large enough: where with stands for the velocity of the incompressible fluid flow and P for the scalar pressure function, which guarantees the divergence free condition of the velocity field, with , and denotes the vertical viscous coefficient.
When , is exactly the classical Navier-Stokes system. In the sequel, we shall always denote the system by . Whereas when , reduces to the anisotropic Navier-Stokes system arising from geophysical fluid mechanics (see [5]). The main motivation for us to study Navier-Stokes system with large vertical viscous coefficient comes from the study of Navier-Stokes system on thin domains (see (2.4) of [26] for instance), which we shall present more details later on.
In the seminal paper [22], Leray proved the global existence of finite energy weak solutions to . Yet the uniqueness and regularity of such weak solutions are big open questions in the field of mathematical fluid mechanics except the case when the initial data have special structure. For instance, with axi-symmetric initial velocity and without swirl component, Ladyzhenskaya [21] and independently Ukhovskii and Yudovich [30] proved the existence of weak solution along with the uniqueness and regularity of such solution to . When the initial data has a slow space variable, Chemin and Gallagher [6] (see also [8]) can also prove the global well-posedness of such a system.
Furthermore, the classical Ladyzhenskaya-Prodi-Serrin regularity criterion claims that Leray solutions of is regular and unique as long as with satisfying and . The end-point case for was solved by Iskauriaza, Serëgin and Šverák in [18]. Beirão da Veiga [2] proved that the local smooth solution of will not blow up at time T if v satisfies with satisfying for . We remark that the study of geometric criterion for the regularity of was pioneered by Constantin in [10] where an explicit representation formula for the stretching factor in the evolution of the vorticity magnitude was derived. This geometric depletion of the nonlinearity was utilized by Constantin and Fefferman in [11] to prove that if the direction of vorticity is sufficiently well behaved in regions of high vorticity magnitude, then the solution is smooth. A full spatiotemporal localization of the vorticity formulation of was developed by Dascaliuc, Grujić and Guberović. One may check [12], [14], [15], [16] and the references therein for details.
While Fujita and Kato [13] proved the global well-posedness of when the initial data is sufficiently small in the homogeneous Sobolev space . This result was generalized by Cannone, Meyer and Planchon [4] for initial data being sufficiently small in the homogeneous Besov space, , with . The end-point result in this direction is due to Koch and Tataru [19], where they proved the global well-posedness of with initial data being sufficiently small in , the norm of which is determined by We remark that for , there holds and the norms to the above spaces are sclaing-invariant under the following scaling transformation We notice that for any solution u of on , determined by (1.2) is also a solution of on . We remark that the largest space, which belongs to and the norm of which is scaling invariant under (1.2), is . Moreover, Bourgain and Pavlović [3] proved that is actually ill-posed with initial data in .
On the other hand, by crucially using the fact that , Zhang [31], Paicu and the second author [24] improved Fujita and Kato's result by requiring only two components of the initial velocity being sufficiently small in some critical Besov space even when in . Lately, Chemin and Zhang [9] proved that if the lifespan to the Fujita-Kato solution of is finite, then for any unit vector field e of , and any , there holds This result ensures that a critical norm to one component of the velocity field controls the regularity of Fujita-Kato solution to . In general, we still do not know whether or not is globally well-posed with only one component of the initial velocity being sufficiently small. Yet we shall prove the global well-posedness of with a family of initial data, which vary fast enough in the vertical direction and the third component of which are sufficiently small, see Corollary 1.1 below.
Before preceding, let us recall the anisotropic Sobolev space.
Definition 1.1 For any , denotes the space of homogeneous tempered distribution a such that
Notations Let us denote , and For any function a and any positive constant λ, we denote . designates the vorticity of the velocity v, and , the third component of Ω.
Our first result of this paper states as follows:
Theorem 1.1 Let satisfy and . Then there exists some universal positive constant such that if has a unique global solution so that for any ,
Remark 1.1 Due to , we deduce from Sobolev inequality and Biot-Savart's law that and This implies that under the assumption of Theorem 1.1, determined by (1.4) is well-defined. Furthermore, if is sufficiently small, (1.4) holds for . Hence in particular, Theorem 1.1 ensures the global well-posedness of the classical 3-D Navier-Stokes system with being sufficiently small. We also mention that lately Paicu and the second author of this paper considered the competition between the horizontal and vertical viscous coefficients of the Navier-Stokes system in [25].
We point out that the main idea used to prove Theorem 1.1 can be adapted to study the global well-posedness of the classical 3-D Navier-Stokes equations with a fast variable: Let and . Then verifies (1.7) is closely related to the following system: which is the rescaled Navier-Stokes system (see (2.4) of [26]) arising from the study of 3-D Navier-Stokes system on thin domains, . In [26] (see also [17], [20], [28]), Raugel and Sell proved the global well-posedness of (1.8) in a periodic domain, , provided that ν is sufficiently large compared to the initial data. The main ideas in [26], [17], [20], [28] is to decompose the solution v of (1.8) as Then the authors exploited the fact that: 2-D Navier-Stokes system is globally well-posed for any data in , and the fact that: 3-D Navier-Stokes system is globally well-posedness with small regular initial data, to prove that the solutions of (1.8) can be split as the sum of a 2-D large solution and a 3-D small solution of .
We remark that in the whole space case, we do not know how to define the average of the velocity field on the vertical variable. Thus it is not clear how to apply the ideas in [26], [17], [20], [28] to solve (1.7). Our principle result concerning the well-posedness of the system (1.7) is as follows:
Theorem 1.2 Consider the re-scaled Navier-Stokes system (1.7) with initial data satisfying and . There exists small positive constants , such that if or then the system (1.7) has a unique global solution .
In particular, the above theorem ensures the global well-posedness of (1.6) provided that the profile of the initial data satisfying (1.10) or (1.11). For the special case when vanishes, we have the following direct consequence: Corollary 1.1 For any with being sufficiently small, (1.6) with initial data has a unique global solution provided that ν is so large that for some positive constant .
Remark 1.2 For arbitrary smooth functions and with and , we take . Then the corresponding initial data given by (1.12) reads In particular, let us take f so that Then we can select ν so large that for any , where denotes . Furthermore, by virtue of (1.1), we have Notice that for any satisfying for near 0, and any , we can take and , where ε is some sufficiently small positive constant relying on the choice of and . Then there holds and As a result, we have which is sufficiently small. Thus all the conditions in Corollary 1.1 are satisfied, and this indeed can generate a unique global solution. On the other side, we have so in general can be arbitrarily large. Hence Theorem 1.2 and Corollary 1.1 can not be deduced from the end-point result in [19].
Sketch of the paper. Motivated by [9], we first reformulate as Then due to , given , by Biot-Savart's law, we write
Let us recall the following result from [9]:
Theorem 1.3 Let us consider an initial data with vorticity . Then a unique maximal solution v of exists in the space for some maximal existing time , and this solution satisfies Moreover, if , for any , we have
According to Theorem 1.3, in order to prove Theorem 1.1, it remains to verify that (1.16) can never be satisfied under the assumption (1.4). It is easy to observe that It reduces to derive the estimate for and the estimate of , that is, the estimate of . In view of (1.14) and (1.15), in order to close the estimates, we also need the estimate for ω. As a matter of fact, under the assumption (1.4), we can indeed achieve the estimate (1.5). This in turn shows that is finite for any and any . Then Theorem 1.1 follows from Theorem 1.3.
Along the same line, we can equivalently reformulate the system (1.7) as Obviously, the difference between the systems (1.14) and (1.18) is that there appears powers of ν in the front of the quadric terms in (1.18), which makes it more difficult to perform the uniform estimates. Indeed the most dangerous term is , which is more or less the same as . Thus if we want to close the previous estimates for sufficiently large ν, it seems necessary that ω should be small for all time. That is the reason why we need the smallness condition (1.10) in Theorem 1.2.
Section snippets
Preliminaries
We first recall some basic facts on anisotropic Littlewood-Paley theory from [1]: where , and denote the Fourier transform of the distribution a, and are smooth functions such that
Definition 2.1 Let us define the space (with usual adaptation when
The proof of Theorem 1.1
This section is devoted to the proof of Theorem 1.1. And the strategy is to verify that, the necessary condition for finite time blow-up criteria, (1.16), can never be satisfied for the local Fujita-Kato solutions of under the assumption of (1.4).
Global solution of (1.6) with a fast variable
In this section, we shall adapt the arguments in the previous section to prove Theorem 1.2.
Acknowledgements
P. Zhang would like to thank Professor L. Caffarelli and Professor J.-Y. Chemin for profitable discussions on this topic. The authors would also like to thank the profitable suggestions of the referees.
P. Zhang is partially supported by NSF of China under Grants 11371347 and 11688101, Morningside Center of Mathematics of The Chinese Academy of Sciences and innovation grant from National Center for Mathematics and Interdisciplinary Sciences.
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