A New Convex Integration Approach for the Compressible Euler Equations and Failure of the Local Maximal Dissipation Criterion

Abstract

In this paper we establish a new convex integration approach for the barotropic compressible Euler equations in two space dimensions. In contrast to existing literature, our new method generates not only the momentum for given density, but also the energy and the energy flux. This allows for a simple way to construct admissible solutions, i.e. solutions which satisfy the energy inequality. Moreover using the convex integration method developed in this paper, we show that the local maximal dissipation criterion fails in the following sense: There exist wild solutions which beat the self-similar solution of the one-dimensional Riemann problem extended to two dimensions. Hence the local maximal dissipation criterion rules out the self-similar solution. The convex integration machinery itself is carried out in a very general way. Hence this paper provides a universal framework for convex integration which not even specifies the form of the partial differential equations under consideration. Therefore this general framework is applicable in many different situations.

arXiv comment62 pages

References in corpus (10)

• On admissibility criteria for weak solutions of the Euler equations
• Global ill-posedness of the isentropic system of gas dynamics
• A counterexample to well-posedness of entropy solutions to the compressible Euler system
• On the energy dissipation rate of solutions to the compressible isentropic Euler system
• Maximal dissipation and well-posedness for the compressible Euler system
• The Riemann problem for the multidimensional isentropic system of gas dynamics is ill-posed if it contains a shock
• Non-uniqueness of admissible weak solutions to the Riemann problem for the isentropic Euler equations
• The 2-d isentropic compressible Euler equations may have infinitely many solutions which conserve energy
• On bounded two-dimensional globally dissipative Euler flows
• On contact discontinuities in multi-dimensional isentropic Euler equations

Cited by in corpus (3)

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