Global well-posedness of 3-D anisotropic Navier-Stokes system with large vertical viscous coefficient

https://doi.org/10.1016/j.jfa.2020.108736Get rights and content

Abstract

In this paper, we first prove the global well-posedness of 3-D anisotropic Navier-Stokes system provided that the vertical viscous coefficient of the system is sufficiently large compared to some critical norm of the initial data. Then we shall construct a family of initial data, u0,ν, which vary fast enough in the vertical variable and which can be arbitrarily large in the space BMO1. Yet u0,ν still generates a unique global solution to the classical 3-D Navier-Stokes system provided that ν is sufficiently large.

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:(NSν){tv+vvΔνv+P=0,(t,x)R+×R3,divv=0,v|t=0=v0=(v0h,v03), where v=(vh,v3) with vh=(v1,v2) 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, Δν=defΔh+ν232 with Δh=def12+22, and ν2 denotes the vertical viscous coefficient.

When ν=1, (NSν) is exactly the classical Navier-Stokes system. In the sequel, we shall always denote the system (NS1) by (NS). Whereas when ν=0, (NSν) 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 (NS). 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 (NS). When the initial data v0 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 (NS) is regular and unique as long as vLq(0,T;Lp(R3)) with p,q satisfying 2q+3p=1 and p>3. The end-point case for p=3 was solved by Iskauriaza, Serëgin and Šverák in [18]. Beirão da Veiga [2] proved that the local smooth solution of (NS) will not blow up at time T if v satisfies vLr(0.T;Ls(R3)) with r,s satisfying 2r+3s=2 for 3s<. We remark that the study of geometric criterion for the regularity of (NS) 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 (NS) 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 (NS) when the initial data v0 is sufficiently small in the homogeneous Sobolev space H12. This result was generalized by Cannone, Meyer and Planchon [4] for initial data being sufficiently small in the homogeneous Besov space, Bp,1+3p, with p]3,[. The end-point result in this direction is due to Koch and Tataru [19], where they proved the global well-posedness of (NS) with initial data being sufficiently small in BMO1, the norm of which is determined byvBMO1=defvB,1+supxR3,R>01R32(0R2B(x,R)|etΔv(y)|2dydt)12. We remark that for p]3,[, there holdsH12(R3)L3(R3)Bp,1+3p(R3)BMO1(R3)B,1(R3), and the norms to the above spaces are sclaing-invariant under the following scaling transformationuλ(t,x)=defλu(λ2t,λx)andu0,λ(x)=defλu0(λx). We notice that for any solution u of (NS) on [0,T], uλ determined by (1.2) is also a solution of (NS) on [0,T/λ2]. We remark that the largest space, which belongs to S(R3) and the norm of which is scaling invariant under (1.2), is B,1(R3). Moreover, Bourgain and Pavlović [3] proved that (NS) is actually ill-posed with initial data in B,1.

On the other hand, by crucially using the fact that divv=0, 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 ν=0 in (NSν). Lately, Chemin and Zhang [9] proved that if the lifespan T to the Fujita-Kato solution of (NS) is finite, then for any unit vector field e of R3, and any p]4,6[, there holds0TveH12+2ppdt=. This result ensures that a critical norm to one component of the velocity field controls the regularity of Fujita-Kato solution to (NS). In general, we still do not know whether or not (NS) is globally well-posed with only one component of the initial velocity being sufficiently small. Yet we shall prove the global well-posedness of (NS) 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 s,sR, Hs,s denotes the space of homogeneous tempered distribution a such thataHs,s2=defR3|ξh|2s|ξ3|2s|aˆ(ξ)|2dξ<withξh=(ξ1,ξ2).

Notations

Let us denote h=def(1,2),h=def(2,1), ν=def(1,2,ν3) andET=defC([0,T[,H12)Lloc2([0,T[;H32). For any function a and any positive constant λ, we denote aλ=defa|a|λ1. Ω=defcurlv designates the vorticity of the velocity v, and ω=def1v22v1, the third component of Ω.

Our first result of this paper states as follows:

Theorem 1.1

Let v0 satisfy Ω0=curlv0L32 and divv0=0. Then there exists some universal positive constant C1 such that ifνC1(M0+M014)withM0=defω0L3232+v03H12,02, (NSν) has a unique global solution vE so that for any t>0,ω(t)L3232+v3(t)H12,02+0t(νω34L22+νv3H12,02)dt2M0.

Remark 1.1

Due to divv0=0, we deduce from Sobolev inequality and Biot-Savart's law thatv03H12,0v0H12v0L32Ω0L32, and3v03H12,02=|ξ3||ξh||ξh|1|F(3v03)(ξ)|2dξ+|ξh||ξ3||ξh|1|F(divhv0h)(ξ)|2dξR3(|ξ3||vˆ03(ξ)|2+|ξh||vˆ0h(ξ)|2)dξv0H122Ω0L322. This implies that under the assumption of Theorem 1.1, M0 determined by (1.4) is well-defined. Furthermore, if Ω0L32 is sufficiently small, (1.4) holds for ν=1. Hence in particular, Theorem 1.1 ensures the global well-posedness of the classical 3-D Navier-Stokes system with Ω0L32 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:{tu+uuΔu+Π=0,(t,x)R+×R3,divu=0,u|t=0=v0,ν(xh,νx3)=(νv0h(xh,νx3),v03(xh,νx3)). Let u(t,x)=def(νvh(t,xh,νx3),v3(t,xh,νx3)) and Π(t,x)=defνP(t,xh,νx3). Then (v,P) verifies{tv+νvvΔνv+ν2P=0,(t,x)R+×R3,v=0,v|t=0=v0. (1.7) is closely related to the following system:{tv+vνvΔνv+νP=0,(t,x)R+×[0,L1]×[0,L2]×[0,1],νv=0,v|t=0=v0,ν=(νv0h,v03), 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, [0,L1]×[0,L2]×[0,1/ν]. In [26] (see also [17], [20], [28]), Raugel and Sell proved the global well-posedness of (1.8) in a periodic domain, [0,L1]×[0,L2]×[0,1], 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) asv=M(v)+wwithM(v)(t,xh)=def01v(t,xh,x3)dx3. Then the authors exploited the fact that: 2-D Navier-Stokes system is globally well-posed for any data in L2, 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 (NS).

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 v0 satisfying divv0=0 and Ω0L32. There exists small positive constants c1,c2, such that ifω0L32c1ν23hv03L3212,ν23hv03L32c2and3v03L32hv03L3212c1c2,c123ν1hv03L323v03L32, orω0L32c1ν23hv03L3212,ν23hv03L32c2,andv03L32hv03L3212c1c2, then the system (1.7) has a unique global solution vE.

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 ω0 vanishes, we have the following direct consequence:

Corollary 1.1

For any φW1,32 with φL322hφL32 being sufficiently small, (1.6) with initial datau0,ν(x)=(νhΔh13φ,φ)(xh,νx3), has a unique global solution provided that ν is so large thatνC2hφL3232 for some positive constant C2.

Remark 1.2

For arbitrary smooth functions f(xh) and g(x3) with hfS(R2) and 3gS(R), we take φ(x)=defΔhf(xh)g(x3). Then the corresponding initial data given by (1.12) readsu0,ν(x)=(νhf(xh)3g(νx3),Δhf(xh)g(νx3)). In particular, let us take f so that|hf(xh)|2 in some small neighborhood ofxh=0. Then we can select ν so large that |etΔhhf(xh)|1 for any (t,xh)Pν1, where PR denotes [0,R2]×BR. Furthermore, by virtue of (1.1), we haveu0,νhBMO1ν32(Pν1|etΔ(νhf(xh)3g(νx3))|2dxdt)12=ν52(Pν1|(etΔhhf)(xh)(eν2t323g)(νx3)|2dxdt)12Cν32(0ν212ν112ν1|(eν2t323g)(νx3)|2dx3dt)12=Cet323gL2([0,1]×[12,12]). Notice that for any χ1Cc(R2) satisfying |χ1(xh)|2 for xh near 0, and any χ2Cc(R), we can take hf(xh)=χ1(εxh) and g(x3)=ε16χ2(ε1x3), where ε is some sufficiently small positive constant relying on the choice of χ1 and χ2. Then there holdshφL32(R3)=ε2(Δhχ1)(εxh)L32(R2)ε16χ2(ε1x3)L32(R)ε32, and3φL32(R3)=ε(divhχ1)(εxh)L32(R2)ε56(3χ2)(ε1x3)L32(R)ε12. As a result, we haveφL32(R3)2hφL32(R3)=hφL32(R3)3+3φL32(R3)2hφL32(R3)ε12, which is sufficiently small. Thus all the conditions in Corollary 1.1 are satisfied, and this u0,ν indeed can generate a unique global solution. On the other side, we have3gL2([14,14])ε13, so u0,νhBMO1 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 (NSν) as{tω+vωΔνω=3v3ω+2v33v11v33v2,tv3+vv3Δνv3=3Δ1(,m=13vmmv),ω|t=0=ω0,v3|t=0=v03. Then due to divhvh=3v3, given (ω,v3), by Biot-Savart's law, we writevh=vcurlh+vdivh,wherevcurlh=defhΔh1ωandvdivh=defhΔh13v3.

Let us recall the following result from [9]:

Theorem 1.3

Let us consider an initial data v0 with vorticity Ω0L32. Then a unique maximal solution v of (NSν) exists in the space ET for some maximal existing time T>0, and this solution satisfiesΩ=defcurlvC([0,T[,L32)and|Ω||Ω|14Lloc2([0,T[;L2) Moreover, if T<, for any p]4,6[, we have0Tv3(t)H12+2ppdt=.

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 thatv3H12+2pv3H12+2p,0+v3H0,12+2pv3H12,012phv3H12,02p+3v3H12,0122pv3H12,01232v3H12,02p. It reduces to derive the H12,0 estimate for v3 and the H12,0 estimate of 3v3, that is, the H12,0 estimate of v3. In view of (1.14) and (1.15), in order to close the estimates, we also need the L32 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 v3LTp(H12+2p) is finite for any p]4,6[ and any T<. Then Theorem 1.1 follows from Theorem 1.3.

Along the same line, we can equivalently reformulate the system (1.7) as{tω+νvωΔνω=ν(3v3ω+2v33v11v33v2),tv3+νvv3Δνv3=ν23Δ1(,m=13νvmmv),ω|t=0=ω0,v3|t=0=v03. 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 ν23Δ1(,m=12νvcurlmmvcurl), which is more or less the same as ν33Δ1(ω2). 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]:Δja=F1(φ(2j|ξ|)aˆ),Δkha=F1(φ(2k|ξh|)aˆ),Δva=F1(φ(2|ξ3|)aˆ),Sja=F1(χ(2j|ξ|)aˆ),Skha=F1(χ(2k|ξh|)aˆ),Sva=F1(χ(2|ξ3|)aˆ), where ξh=(ξ1,ξ2), Fa and aˆ denote the Fourier transform of the distribution a, χ(τ) and φ(τ) are smooth functions such thatSuppφ{τR/34|τ|83}andτ>0,jZφ(2jτ)=1,Suppχ{τR/|τ|43}andχ(τ)+j0φ(2jτ)=1.

Definition 2.1

Let us define the space (Bp,q1s1)h(Bp,q2s2)v (with usual adaptation when q1

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 (NSν) 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.

References (31)

  • J.-Y. Chemin et al.

    Remarks on the global solutions of 3-D Navier-Stokes system with one slow variable

    Commun. Partial Differ. Equ.

    (2015)
  • J.-Y. Chemin et al.

    On the critical one component regularity for 3-D Navier-Stokes system

    Ann. Sci. Éc. Norm. Supér. (4)

    (2016)
  • P. Constantin

    Geometric statistics in turbulence

    SIAM Rev.

    (1994)
  • P. Constantin et al.

    Direction of vorticity and the problem of global regularity for the Navier-Stokes equations

    Indiana Univ. Math. J.

    (1993)
  • R. Dascaliuc et al.

    Coherent vortex structures and 3D enstrophy cascade

    Commun. Math. Phys.

    (2013)
  • Cited by (0)

    View full text