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

2013/07/26 by Bourgain, Jean, Li, Dong
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1307.7090

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^∞-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.

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