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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

TL;DR

The paper identifies sharp low-regularity ill-posedness thresholds for the 2D ideal compressible MHD system and, in the zero-magnetic-field limit, the 2D Euler equations. It combines a wave-decomposition method with planar symmetry and carefully designed log-type initial data to produce instantaneous shock formation, causing the lifespan to vanish and Sobolev norms to inflate. The main contributions are isotropic ill-posedness in $H^s$ for $s<7/4$ and anisotropic ill-posedness at $s=7/4$, valid for both strictly and non-strictly hyperbolic regimes, with Euler recovered as a corollary. This work sharpens the understanding of regularity thresholds in multi-wave speed hyperbolic systems and provides the first explicit low-regularity ill-posedness results for the 2D ideal compressible MHD system.

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.

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

TL;DR

The paper identifies sharp low-regularity ill-posedness thresholds for the 2D ideal compressible MHD system and, in the zero-magnetic-field limit, the 2D Euler equations. It combines a wave-decomposition method with planar symmetry and carefully designed log-type initial data to produce instantaneous shock formation, causing the lifespan to vanish and Sobolev norms to inflate. The main contributions are isotropic ill-posedness in for and anisotropic ill-posedness at , valid for both strictly and non-strictly hyperbolic regimes, with Euler recovered as a corollary. This work sharpens the understanding of regularity thresholds in multi-wave speed hyperbolic systems and provides the first explicit low-regularity ill-posedness results for the 2D ideal compressible MHD system.

Abstract

In a fractional Sobolev space with , 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 regularity threshold for the 2D compressible Euler system with respect to the fluid velocity and density.
Paper Structure (19 sections, 5 theorems, 188 equations, 6 figures)

This paper contains 19 sections, 5 theorems, 188 equations, 6 figures.

Table of Contents

  1. Introduction and Main results
  2. Background and history
  3. Main results
  4. Organization of the paper
  5. Acknowledgements
  6. Preliminaries
  7. The corresponding hyperbolic system
  8. Decomposition of waves
  9. Boundedness of the coefficients
  10. Characteristic strips
  11. Constructions of initial data
  12. Shock formation
  13. $L^\infty$ estimates
  14. Finite-time blow-up of $w_1$
  15. Upper bound estimates for other $w_i$ ($i\neq 1$)
  16. ...and 4 more sections

Key Result

Theorem 1.1

We say that the Cauchy problem of the 2D ideal compressible MHD equations MHD is ill-posed in $H^s(\mathbb{R}^2)$ if: There exists a family of compactly supported, smooth initial data satisfying with $\kappa$ denoting the constant for equilibrium state, and $\eta>0$ being a small parameter. For each initial datum, there exists finite $T_\eta^*>0$ such that the corresponding Cauchy problem of the

Figures (6)

  • Figure 1: Vorticity regularity $s'$vs. density regularity $\tilde{s}$.
  • Figure 2: Vorticity regularity $s'$vs. velocity regularity $s$.
  • Figure 3:
  • Figure 4: Separation of five characteristic strips.
  • Figure 5: Domain of dependence. In this picture, the domain of dependence $\mathcal{D}(\Omega_0;T_\eta^*)$ corresponds to the whole colored trapezoid region. According to the finite speed of propagation, within $\mathcal{D}(\Omega_0;T_\eta^*)$, the solutions, as well as the initial data, are compactly supported in the dark gray region. While, the light gray region denotes the trivial part. Hence the dashed line in this figure is straight.
  • ...and 1 more figures

Theorems & Definitions (23)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.2
  • Remark 1.6
  • Remark 2.1
  • Lemma 3.1
  • ...and 13 more