Local ill-posedness of the Euler equations in $B^1_{\infty,1}$

Gerard Misiołek; Tsuyoshi Yoneda

Local ill-posedness of the Euler equations in $B^1_{\infty,1}$

Analysis of PDEs 2016-03-27 v1

Authors: Gerard Misiołek , Tsuyoshi Yoneda

Abstract

We show that the incompressible Euler equations on $\mathbb{R}^2$ are not locally well-posed in the sense of Hadamard in the Besov space $B^1_{\infty,1}$ . Our approach relies on the technique of Lagrangian deformations of Bourgain and Li. We show that the assumption that the data-to-solution map is continuous in $B^1_{\infty,1}$ leads to a contradiction with a well-posedness result in $W^{1,p}$ of Kato and Ponce.

Keywords

euler equations ill-posedness besov space hadamard well-posedness lagrangian deformations data-to-solution map

Cite

@article{arxiv.1405.4933,
  title  = {Local ill-posedness of the Euler equations in $B^1_{\infty,1}$},
  author = {Gerard Misiołek and Tsuyoshi Yoneda},
  journal= {arXiv preprint arXiv:1405.4933},
  year   = {2016}
}

Comments

arXiv admin note: substantial text overlap with arXiv:1405.1943

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