The Cauchy problems for the 2D compressible Euler equations and ideal MHD system are ill-posed in $H^\frac{7}{4}(\mathbb{R}^2)$

Xinliang An; Haoyang Chen; Silu Yin

The Cauchy problems for the 2D compressible Euler equations and ideal MHD system are ill-posed in $H^\frac{7}{4}(\mathbb{R}^2)$

Analysis of PDEs 2026-01-27 v3 Mathematical Physics math.MP Fluid Dynamics

Authors: Xinliang An , Haoyang Chen , Silu Yin

Abstract

In a fractional Sobolev space $H^s(\mathbb{R}^2)$ with $s\leq\frac74$ , we prove the low-regularity ill-posedness for the 2D compressible Euler equations and the 2D ideal compressible MHD system. Our ill-posedness results match the $H^\frac74$ regularity threshold for the 2D compressible Euler system with respect to the fluid velocity and density.

Keywords

compressible euler equations ideal compressible mhd ill-posedness low regularity fractional sobolev space cauchy problem

Cite

@article{arxiv.2206.14003,
  title  = {The Cauchy problems for the 2D compressible Euler equations and ideal MHD system are ill-posed in $H^\frac{7}{4}(\mathbb{R}^2)$},
  author = {Xinliang An and Haoyang Chen and Silu Yin},
  journal= {arXiv preprint arXiv:2206.14003},
  year   = {2026}
}

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