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

Abstract

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

119 pages

References in corpus (1)

• On the well-posedness of the incompressible Euler Equation

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• A mathematical consideration of vortex thinning in 2D turbulence
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