Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces

By Jean Bourgain, Dong Li. Inventiones Mathematicae, 2013

203 citations recorded by the source corpus.

For the d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d$$\end{document}-dimensional incompressible Euler equation, the standard energy method gives local wellposedness for initial velocity in Sobolev space Hs(Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^s(\mathbb R^d)$$\end{document}, s>sc:=d/2+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s>s_c:=d/2+1$$\end{document}. The borderline case s=sc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=s_c$$\end{document} was a folklore open problem. In this paper we consider the physical dimension d=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=2$$\end{document} and show that if we perturb any given smooth initial data in Hsc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{s_c}$$\end{document} norm, then the corresponding solution can have infinite Hsc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{s_c}$$\end{document} norm instantaneously at t>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t>0$$\end{document}. In a companion paper [1] we settle the 3D and more general cases. The constructed solutions are unique and even C∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{\infty }$$\end{document}-smooth in some cases. To prove these results we introduce a new strategy: large Lagrangian deformation induces critical norm inflation. As an application we also settle several closely related open problems.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/440c8fa467c5eb6f29253c85a86426f94af10f98

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Citations in the corpus

On Kato–Ponce and fractional Leibniz (2016, background): “Further counterexamples In this section we collect further counterexamples for the operator J on divergence-free velocity fields which are deeply connected with the investigation of norm estimates of incompressible Euler equations in Sobolev spaces (see [2, 3, 17] and the references therein).”
L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty $$\end{document} Ill-Posedness for a Class of Equations Ari (2019, background): “A very interesting work byBourgain and Li [6] studies the ill-posedness of the Euler equations with velocity in H d 2 +1, the difficulty being that H d 2 +1 is a critical space sitting at the lower threshold of the classes of strong solutions where local well-posedness holds (the Euler equations are”
Strong illposedness of the incompressible Euler equation in integer Cm spaces (2014, background): “1,1! This is perhaps a bit surprising since we do have wellposedness in C1,α for any 0 < α < 1. To prove the above theorem, a very tempting thought is to follow directly the scheme developed in [5] and prove the inﬂation through large Lagrangian deformation with high frequency perturbation. As”
Convex integration and phenomenologies in turbulence (2019, background): “Unfortunately, it is not known if this procedure is valid exactly in the presence of strong anisotropies such as occur [10] in pipe flows, or if some nontrivial correction is needed [11].”
Ill-posedness of the Camassa–Holm and related equations in the critical space (2018, result): “ace H3/2 (non-continuity of the ﬂow map at 0). The norm inﬂation is even stronger than the classical sense, namely weak norm ku(t)kB1 ∞,∞(K) blows up. Recently, for Euler equation, Bourgain and Li in [1, 2] proved strong local illposedness in borderline Besov spaces Bd/p+1 p,r for 1 ≤ p < ∞ and 1”
Local Existence for the Non-Resistive MHD Equations in Nearly Optimal Sobolev Spaces (2016, background): “Bourgain and Li [3] showed that the Euler equations on Rd are ill posed in H1+d/2 for n = 2, 3, and we have shown via an explicit example that for the heat equation we cannot gain the time integrability of two additional derivatives that is required in our local existence argument.”
Higher order commutator estimates and local existence for the non-resistive MHD equations and related models (2014, unknown): “We note that in a recent paper Bourgain & Li (2013) showed that the Euler equations on Rn are in fact ill-posed in H1+n/2 (n = 2, 3); in light of this it seems likely that system (1.1) is ill-posed in Hn/2.”
Global solutions of 3D incompressible MHD system with mixed partial dissipation and magnetic diffusion near an equilibrium (2019, methodology): “The 3D Navier-Stokes equations with full dissipation have small data global well-posedness while the 3D incompressible Euler equations are ill-posed and have norm inflation in any Sobolev space H or C for any positive integer k [4,5,18].”