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Non-uniqueness of admissible weak solutions to the two-dimensional barotropic compressible Euler system with contact discontinuities

Kotaro Horimoto

Abstract

This paper is concerned with the Riemann problem for the two-dimensional barotropic compressible Euler system with a general strictly increasing pressure law. By means of convex integration, the existence of infinitely many admissible weak solutions is established for certain Riemann initial data for which the corresponding one-dimensional self-similar solution consists solely of a contact discontinuity.

Non-uniqueness of admissible weak solutions to the two-dimensional barotropic compressible Euler system with contact discontinuities

Abstract

This paper is concerned with the Riemann problem for the two-dimensional barotropic compressible Euler system with a general strictly increasing pressure law. By means of convex integration, the existence of infinitely many admissible weak solutions is established for certain Riemann initial data for which the corresponding one-dimensional self-similar solution consists solely of a contact discontinuity.
Paper Structure (5 sections, 6 theorems, 59 equations)

This paper contains 5 sections, 6 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions
  4. Sufficient condition for non-uniqueness
  5. Proof of \ref{['thm:main']}

Key Result

Theorem 1.1

Assume that $p$ satisfies eq:p-condition. Let $\rho_0>0$ and $u_0\neq 0$ be arbitrarily given. Set $\rho_+ = \rho_- = \rho_0$, $\bm{m}_+=(\rho_0u_0 \,,\, 0)$, and $\bm{m}_-=(-\rho_0u_0 \,,\, 0)$. Then there exist infinitely many admissible weak solutions to eq:Euler mass--eq:Euler init2 with eq:Riem

Theorems & Definitions (20)

  • Definition 1: Weak solutions
  • Definition 2: Admissible weak solutions
  • Theorem 1.1
  • Remark 1
  • Definition 3: Fan partition, see Mark2024
  • Remark 2
  • Definition 4: Admissible fan subsolution, see Mark2024
  • Remark 3
  • Proposition 1: cf. Mark2024
  • Remark 4
  • ...and 10 more