Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Contact discontinuities for 2-D isentropic Euler are unique in 1-D but wildly non-unique otherwise

Sam G. Krupa, László Székelyhidi

TL;DR

We address non-uniqueness of contact discontinuities for the 2-D isentropic Euler system by exploiting convex integration with the pressure law $p$ treated as a free, strictly increasing function ($p'>0$). The authors construct admissible fan subsolutions and, aided by a computer-assisted algebraic solver and a quantified inverse function theorem, produce infinitely many bounded, admissible weak solutions for Riemann data where $\rho_+=\rho_-$ and $v_{+2}=v_{-2}$ but $v_{+1}\neq v_{-1}$; they also show that 1-D solutions (depending only on $x_2$) are unique in the same class, via a switch to Lagrangian coordinates and the weak-strong stability framework. This work delineates the extent of non-uniqueness in multi-D Riemann problems for isentropic Euler and clarifies conditions under which dimensionality enforces uniqueness, with implications for understanding stability and limits of vanishing-viscosity models. The approach combines convex integration theory, rigorous subsolution construction, and computer-assisted proofs to establish these sharp results.

Abstract

We develop a general framework for studying non-uniqueness of the Riemann problem for the isentropic compressible Euler system in two spatial dimensions, and in this paper we present the most delicate result of our method: non-uniqueness of the contact discontinuity. Our approach is computational, and uses the pressure law as an additional degree of freedom. The stability of the contact discontinuities for this system is a major open problem (see Gui-Qiang Chen and Ya-Guang Wang [Nonlinear partial differential equations, volume 7 of Abel Symposia. Springer, Heidelberg, 2012.]). We find a smooth pressure law $p$, verifying the physically relevant condition $p'>0$, such that for the isentropic compressible Euler system with this pressure law, contact discontinuity initial data is wildly non-unique in the class of bounded, admissible weak solutions. This result resolves the question of uniqueness for contact discontinuity solutions in the compressible regime. Moreover, in the same regularity class in which we have non-uniqueness of the contact discontinuity, i.e. $L^\infty$, with no $BV$ regularity or self-similarity, we show that the classical contact discontinuity solution to the two-dimensional isentropic compressible Euler system is in fact unique in the class of bounded, admissible weak solutions if we restrict to 1-D solutions.

Contact discontinuities for 2-D isentropic Euler are unique in 1-D but wildly non-unique otherwise

TL;DR

We address non-uniqueness of contact discontinuities for the 2-D isentropic Euler system by exploiting convex integration with the pressure law treated as a free, strictly increasing function (). The authors construct admissible fan subsolutions and, aided by a computer-assisted algebraic solver and a quantified inverse function theorem, produce infinitely many bounded, admissible weak solutions for Riemann data where and but ; they also show that 1-D solutions (depending only on ) are unique in the same class, via a switch to Lagrangian coordinates and the weak-strong stability framework. This work delineates the extent of non-uniqueness in multi-D Riemann problems for isentropic Euler and clarifies conditions under which dimensionality enforces uniqueness, with implications for understanding stability and limits of vanishing-viscosity models. The approach combines convex integration theory, rigorous subsolution construction, and computer-assisted proofs to establish these sharp results.

Abstract

We develop a general framework for studying non-uniqueness of the Riemann problem for the isentropic compressible Euler system in two spatial dimensions, and in this paper we present the most delicate result of our method: non-uniqueness of the contact discontinuity. Our approach is computational, and uses the pressure law as an additional degree of freedom. The stability of the contact discontinuities for this system is a major open problem (see Gui-Qiang Chen and Ya-Guang Wang [Nonlinear partial differential equations, volume 7 of Abel Symposia. Springer, Heidelberg, 2012.]). We find a smooth pressure law , verifying the physically relevant condition , such that for the isentropic compressible Euler system with this pressure law, contact discontinuity initial data is wildly non-unique in the class of bounded, admissible weak solutions. This result resolves the question of uniqueness for contact discontinuity solutions in the compressible regime. Moreover, in the same regularity class in which we have non-uniqueness of the contact discontinuity, i.e. , with no regularity or self-similarity, we show that the classical contact discontinuity solution to the two-dimensional isentropic compressible Euler system is in fact unique in the class of bounded, admissible weak solutions if we restrict to 1-D solutions.
Paper Structure (14 sections, 8 theorems, 69 equations, 5 figures)

This paper contains 14 sections, 8 theorems, 69 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Some comments on our proofs
  3. \ref{['main_theorem_nonunique']}: Non-uniqueness
  4. Computational experiments
  5. \ref{['main_theorem_unique']}: Uniqueness
  6. Plan for the paper
  7. The engine of convex integration
  8. Proof of \ref{['main_theorem_nonunique']}
  9. Proof of \ref{['MATLAB_prop']}
  10. Proof of \ref{['main_theorem_unique']}
  11. A quantitative inverse function theorem
  12. Explicit values for \ref{['MATLAB_prop']}
  13. Exact values
  14. Approximate values

Key Result

Theorem 1.1

There exists a smooth pressure function $p\colon (0,\infty)\to\mathbb{R}$ such that $p,p'>0$, and the system system-system_adm with this pressure law admits initial data in the form init_data with $\rho_+=\rho_-$, $v_{+ 2}=v_{- 2}$ and $v_{+ 1}\neq v_{- 1}$ which gives rise to infinitely many bounde

Figures (5)

  • Figure 1: A diagram of the contact discontinuity initial data we consider in this paper.
  • Figure 2: A diagram of a potential 1-D subsolution in space-time (see \ref{['subsolution_system']}). In between the left and right interfaces, $(\rho, v,R)$ is a subsolution and $R$ must be positive semi-definite. We also have $\rho=\rho_+=\rho_-=\mathrm{const}$.
  • Figure 3: A diagram of a fan partition in space-time.
  • Figure 4: The fan partition used for our fan subsolution with contact discontinuity initial data, with the slopes of the lines determined by \ref{['MATLAB_prop']}.
  • Figure 5: A schematic of the proof of \ref{['IFT_prop']}.

Theorems & Definitions (15)

  • Theorem 1.1: Non-uniqueness for contact discontinuity initial data
  • Theorem 1.2: Uniqueness of 1-D solutions
  • Conjecture 1.3
  • Definition 2.1: Fan Partition multi_d_illposed
  • Definition 2.2: Fan Subsolutions multi_d_illposed
  • Definition 2.3: Admissible Fan Subsolutions multi_d_illposed
  • Remark 1
  • Proposition 2.4: Reduction to Admissible Fan Subsolutions multi_d_illposed
  • Proposition 2.5: multi_d_illposed and MR3831840
  • Lemma 3.1: The MATLAB Lemma
  • ...and 5 more