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arXiv · 1405.2847

Strong illposedness of the incompressible Euler equation in integer $C^m$ spaces

Jean Bourgain·Dong Li
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Abstract

We consider the $d$-dimensional incompressible Euler equations. We show strong illposedness of velocity in any $C^m$ spaces whenever $m\ge 1$ is an \emph{integer}. More precisely, we show for a set of initial data dense in the $C^m$ topology, the corresponding solutions lose $C^m$ regularity instantaneously in time. In the $C^1$ case, our proof is based on an anisotropic Lagrangian deformation and a short-time flow expansion. In the $C^m$, $m\ge 2$ case, we introduce a flow decoupling method which allows to tame the nonlinear flow almost as a passive transport. The proofs also cover illposedness in Lipschitz spaces $C^{m-1,1}$ whenever $m\ge 1$ is an integer.

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Jean Bourgain, Dong Li. 2014-05-12. Strong illposedness of the incompressible Euler equation in integer $C^m$ spaces. https://doi.org/10.1007/s00222-014-0548-6

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