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

For the $$d$$d-dimensional incompressible Euler equation, the standard energy method gives local wellposedness for initial velocity in Sobolev space $$H^s(\mathbb R^d)$$Hs(Rd), $$s>s_c:=d/2+1$$s>sc:=d/2+1. The borderline case $$s=s_c$$s=sc was a folklore open problem. In this paper we consider the physical dimension $$d=2$$d=2 and show that if we perturb any given smooth initial data in $$H^{s_c}$$Hsc norm, then the corresponding solution can have infinite $$H^{s_c}$$Hsc norm instantaneously at $$t>0$$t>0. In a companion paper [1] we settle the 3D and more general cases. The constructed solutions are unique and even $$C^{\infty }$$C∞-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.

[1]  Akira Ogawa,et al.  Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics , 2002 .

[2]  Tosio Kato,et al.  The Cauchy problem for quasi-linear symmetric hyperbolic systems , 1975 .

[3]  Dongho Chae,et al.  Local existence and blow‐up criterion for the Euler equations in the Besov spaces , 2004 .

[4]  A. Majda,et al.  Vorticity and incompressible flow , 2001 .

[5]  Claude Bardos,et al.  Mathematical Topics in Fluid Mechanics, Volume 1, Incompressible Models , 1998 .

[6]  Tosio Kato.  Nonstationary flows of viscous and ideal fluids in R3 , 1972 .

[7]  A. Majda,et al.  Oscillations and concentrations in weak solutions of the incompressible fluid equations , 1987 .

[8]  Young Ja Park,et al.  Existence of Solution for the Euler Equations in a Critical Besov Space (ℝ n ) , 2004 .

[9]  Edriss S. Titi,et al.  Loss of smoothness and energy conserving rough weak solutions for the 3d Euler equations , 2009, 0906.2029.

[10]  P. Lions.  Mathematical topics in fluid mechanics , 1996 .

[11]  Peter Constantin,et al.  On the Euler equations of incompressible fluids , 2007 .

[12]  Ryo Takada,et al.  Counterexamples of Commutator Estimates in the Besov and the Triebel-Lizorkin Spaces Related to the Euler Equations , 2010, SIAM J. Math. Anal..

[13]  Gebräuchliche Fertigarzneimittel,et al.  V , 1893, Therapielexikon Neurologie.

[14]  H. Inci.  On the well-posedness of the incompressible Euler Equation , 2013, 1301.5997.

[15]  W. Wolibner.  Un theorème sur l'existence du mouvement plan d'un fluide parfait, homogène, incompressible, pendant un temps infiniment long , 1933 .

[16]  Tosio Kato,et al.  Commutator estimates and the euler and navier‐stokes equations , 1988 .

[17]  R. Danchin.  Axisymmetric incompressible flows with bounded vorticity , 2007 .

[18]  J. Marsden,et al.  Groups of diffeomorphisms and the motion of an incompressible fluid , 1970 .

[19]  M. R. Ukhovskii,et al.  Axially symmetric flows of ideal and viscous fluids filling the whole space , 1968 .

[20]  Tosio Kato,et al.  Remarks on the breakdown of smooth solutions for the 3-D Euler equations , 1984 .

[21]  Jean Bourgain,et al.  On an endpoint Kato-Ponce inequality , 2014, Differential and Integral Equations.

[22]  Haim Brezis,et al.  Remarks on the Euler equation , 1974 .

[23]  T. Yanagisawa,et al.  Note on global existence for axially symmetric solutions of the Euler system , 1994 .

[24]  V. I. Yudovich,et al.  Uniqueness Theorem for the Basic Nonstationary Problem in the Dynamics of an Ideal Incompressible Fluid , 1995 .

[25]  Taoufik Hmidi,et al.  On the global well-posedness for the axisymmetric Euler equations , 2008, 0801.2316.

[26]  A. Alexandrou Himonas,et al.  Non-Uniform Dependence on Initial Data of Solutions to the Euler Equations of Hydrodynamics , 2010 .

[27]  Jean-Yves Chemin,et al.  Perfect Incompressible Fluids , 1998 .

[28]  G. Misiołek,et al.  ILL-POSEDNESS EXAMPLES FOR THE QUASI-GEOSTROPHIC AND THE EULER EQUATIONS , 2012 .

[29]  V. I. Yudovich,et al.  Non-stationary flow of an ideal incompressible liquid , 1963 .

[30]  L. Grafakos,et al.  A remark on an endpoint Kato-Ponce inequality , 2013, Differential and Integral Equations.

[31]  M. Vishik,et al.  Hydrodynamics in Besov Spaces , 1998 .

[32]  R. Shvydkoy,et al.  ILL-POSEDNESS OF THE BASIC EQUATIONS OF FLUID DYNAMICS IN BESOV SPACES , 2009, 0904.2196.

[33]  D. Chae,et al.  Logarithmically regularized inviscid models in borderline sobolev spaces , 2012 .