[Paper Review] On contact discontinuities in multi-dimensional isentropic Euler equations

Jan Březina, Elisabetta Chiodaroli+1 more|arXiv (Cornell University)|Jul 3, 2017

Navier-Stokes equation solutions10 references3 citations

TL;DR

This paper establishes the nonuniqueness of bounded admissible weak solutions for the 2D compressible isentropic Euler equations with Riemann initial data that include a contact discontinuity. By constructing a fan subsolution via convex integration techniques and verifying entropy and admissibility conditions, the authors prove the existence of infinitely many solutions when the initial velocity components across the discontinuity satisfy specific inequalities, extending prior nonuniqueness results to cases involving contact discontinuities.

ABSTRACT

In this short note we partially extend the recent nonuniqueness results on admissible weak solutions to the Riemann problem for the 2D compressible isentropic Euler equations. We prove nonuniqueness of admissible weak solutions that start from the Riemann initial data allowing a contact discontinuity to emerge.

Motivation & Objective

• To extend nonuniqueness results for the 2D compressible isentropic Euler equations to Riemann initial data that include a contact discontinuity.
• To investigate whether admissible weak solutions remain unique when the self-similar solution contains a contact discontinuity, particularly when $ v_{-1} \neq v_{+1} $.
• To establish the existence of infinitely many bounded admissible weak solutions under specific velocity and density conditions using subsolution techniques.
• To verify that the constructed solutions satisfy entropy inequality and admissibility conditions, ensuring physical relevance.

Proposed method

• Construct a fan subsolution with piecewise constant states separated by two interfaces, using a self-similar ansatz with velocity $ \beta $ in the middle region.
• Apply the Rankine-Hugoniot conditions across the left and right interfaces to relate states across discontinuities.
• Impose subsolution conditions $ \varepsilon_1 > 0 $, $ \varepsilon_2 > 0 $ to ensure the subsolution lies strictly within the convex hull of admissible states.
• Verify admissibility conditions on both interfaces using inequalities involving pressure, density, and velocity differences.
• Ensure the interface speeds satisfy $ \nu_- < \beta < \nu_+ $, which is confirmed via estimates from prior lemmas in [4].
• Leverage convex integration techniques and the Baire category method, adapted from incompressible Euler theory, to generate infinitely many solutions from a single subsolution.

Experimental results

Research questions

• RQ1 Does the nonuniqueness of admissible weak solutions in the 2D isentropic Euler equations persist when the self-similar solution includes a contact discontinuity?
• RQ2 Can infinitely many bounded admissible weak solutions be constructed for Riemann data with $ v_{-1} \neq v_{+1} $, where a contact discontinuity emerges?
• RQ3 Are the admissibility and entropy inequality conditions satisfied by the constructed subsolutions, ensuring physical relevance?
• RQ4 How do the velocity differences $ v_{-2} - v_{+2} $ and density ratios affect the existence of such solutions?

Key findings

• Infinitely many bounded admissible weak solutions exist for the 2D isentropic Euler equations when the Riemann initial data include a contact discontinuity and $ v_{-1} \neq v_{+1} $.
• The nonuniqueness result is established via a fan subsolution construction that satisfies all Rankine-Hugoniot, subsolution, and admissibility conditions.
• The interface speeds satisfy $ \nu_- < \beta < \nu_+ $, ensuring the subsolution structure is valid and the solution is physically admissible.
• The admissibility conditions on both interfaces are verified using inequalities involving pressure, density, and velocity differences, with $ \varepsilon_1, \varepsilon_2 > 0 $.
• The result extends prior nonuniqueness theorems from [4] and [5] to cases involving contact discontinuities, filling a gap in the literature.
• The existence of such solutions implies that the entropy inequality alone is insufficient to select a unique solution, even in the presence of contact discontinuities.