Elsevier

Journal of Differential Equations

Volume 273, 5 February 2021, Pages 40-57
Journal of Differential Equations

Non-uniform dependence for Euler equations in Besov spaces

https://doi.org/10.1016/j.jde.2020.11.039Get rights and content

Abstract

We prove the non-uniform continuity of the data-to-solution map of the incompressible Euler equations in Besov spaces, Bp,qs, where the parameters p,q and s considered here are such that the local existence and uniqueness result holds.

Introduction

We consider the Cauchy problem governing the motion of a non-viscous and incompressible fluid in a domain ΩRd,d2tu+(u)u+p=0,divu=0,u(0,x)=u0(x),xΩ, where u:R×ΩRd is the velocity field, p:R×ΩR is the pressure function and u0:ΩRd is the divergence free initial velocity.

A Cauchy problem is said to be well-posed in the sense of Hadamard if given an initial data one can show the existence of a unique solution that depends continuously on the initial data. Our main concern here is with the local, in time, theory. We first generalize the periodic result of Himonas and Misiołek [12] and show that continuous dependence on the initial data in the Eulerian coordinates is the best result one can expect for Besov spaces on the torus Td.

Theorem 1

Let sR, 1p,q and d=2,3. The data-to-solution map of the incompressible Euler equations (1) is not uniformly continuous from the unit ball in Bp,qs(Td) into C([0,T],Bp,qs(Td,Rd)).

Liu and Tang [15] extended the periodic result in [12] to B2,rs(Td) for 1r. We further extend their result to include the cases p2 and use similar scales as in [15] for the non-periodic case. Recently, in the special case p=q=, Misiołek and Yoneda [21] have shown that the solution map for the Euler equations is not even continuous in the space of Hölder continuous functions, and thus not locally Hadamard well-posed in B,1+σC1,σ unless restricted to the little Hölder subspace c1,σ. More precisely, they showed that the incompressible Euler equations (1) are locally well-posed in the sense of Hadamard in c1,σ(Rd) for any 0<σ<1 and d=2,3, where c1,σ is the closure of the C functions with respect to the Hölder norm, see [3]. Most recently Bourgain and Li [7] settled the border line case s=0 left open in [12] in the non-periodic case.

As a consequence to Theorem 1 we have the following result.

Corollary 1

The data-to-solution map of the incompressible Euler equations (1) is not uniformly continuous from the unit ball in c1,σ(Td) into C([0,T],c1,σ(Td,Rd)) for any 0<σ<1 and d=2,3.

For the non-periodic case we have the following result.

Theorem 2

Let s>1+d/2, 2q and d=2,3. The data-to-solution map of the incompressible Euler equations (1) is not uniformly continuous from the unit ball in B2,qs(Rd) into C([0,T],B2,qs(Rd,Rd)).

For background on well-posedness of the Euler equations we refer the reader to the monographs of Bahouri, Chemin and Danchin [2], Bertozzi and Majda [4] and Chemin [8]. The first rigorous results in this direction were proved in the framework of Hölder spaces by Gyunter [11], Lichtenstein [20] and Wolibner [29], and subsequently by Kato [16], Yudovich [30], Ebin and Marsden [10] and others. Concerning the properties of the data-to-solution map, the first results can be found in [10] and in the work of Kato [17], who among other things showed that the solution map for Burgers' equation is not Hölder continuous in Hs topology for any Hölder exponent. Further continuity results for the solution map of the Euler equations were obtained in Hs by Kato and Lai [18], and subsequently in Wps by Kato and Ponce [19]. Ever since, the subject has become an active area of research involving many non-linear evolution equations.

It is worth pointing out in passing that Pak and Park [23] established existence and uniqueness of solutions of the Euler equations in B,11 and showed that the solution map is in fact Lipschitz continuous when viewed as a map between B,10 and C([0,];B,10). Later, Cheskidov and Shvydkoy [9] proved that the solution of the Euler equations cannot be continuous as a function of the time variable at t=0 in the spaces Br,s(Td) where s>0 if 2<r and s>d(2/r1) if 1r2. Furthermore, Bourgain and Li [5], [6] showed that for d=2,3 the Euler equations are strongly ill-posed in the Sobolev space Wd/p+1,p for any 1p<, the Besov space Bp,qd/p+1 for any 1p<, 1<q, and in the classical spaces Cm(Rd) and Cm1,1(Rd) for any integer m1, see also [22]. Most recently, Holmes, Keyfitz and Tiglay [13] proved the non-uniform continuity of the solution map for compressible gas in the Sobolev spaces, and Holmes and Tiglay [14] have obtained similar results for the Hunter Saxton equation in Besov spaces.

Section snippets

Preliminaries

In this section we introduce notation and preliminary results. The solutions that we consider take values in the Besov spaces Bp,qs. Given a smooth bump function φ0 supported in the ball of radius 2 and equal to one in the ball of radius 1, we let φ1(ξ)=φ0(ξ/2)φ0(ξ) and set φj(ξ)=φ1(ξ/2j1) for all jN. Any such dyadic partition of unity defines a frequency restriction operator on functions, φj(D), via the relation F[φj(D)f](ξ):=φj(ξ)fˆ(ξ). In what follows we will let X denote either Td or Rd

Non-uniform dependence in Besov spaces - the periodic case

In this section we prove that the solution map is not uniformly continuous in the periodic case.

To proceed we will need the following estimates for high frequencies in Besov spaces.

Lemma 2

Let sR and 1p,q. For any constant aR we have the estimatesnscos(na)Bp,qs(T)1,nssin(na)Bp,qs(T)1,n1.

Proof

We will proceed by cases.

Case I: 1q<. Since any dyadic partition induces an equivalent norm we further assume that the φj's are radial and use the inverse Fourier transform to getcos(na)Bp,qs(

Non-uniform dependence in Hölder spaces - the periodic case

In this section we prove Corollary 1. c1,σ is the space of functions in C1,σB,1+σ satisfying the additional vanishing conditionlimh0sup0<|xy|<h|αu(x)αu(y)||xy|σ=0,|α|=1.

Given that the case p=q= is already included in the scope of Theorem 1 we see that the result holds in C1,σ(Td) where it is expected for as shown in [21], in this case the solution map is not even continuous. Thus our result states that in c1,σ(Td), continuity of the data-to-solution map is optimal. Since c1,σ

Non-uniform dependence in Besov spaces - the non-periodic case

As in Section 3, our general strategy will be to find two sequences of solutions to (7) for which conditions (i)-(iii) are satisfied. We make use of the approximate solutions technique as in [12]. For more details about this technique refer to Tzvetkov [28]. We first select two sequences of bounded approximate solutions, which are arbitrarily close at time zero but separated at later times. We then show that the difference between the approximate solutions and the exact solutions to equation (7)

Acknowledgements

This work was done in full while the author conducted graduate studies in the Department of Mathematics at the University of Notre Dame. The author would like to thank the José Enrique Fernández Fellowship for the support during his years at Notre Dame and Gerard Misiołek for all the conversations that lead to the development of this manuscript.

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