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Publication# On the energy dissipation rate of solutions to the compressible isentropic Euler system

Abstract

In this paper we extend and complement the results in [4] on the well-posedness issue for weak solutions of the compressible isentropic Euler system in 2 space dimensions with pressure law $p(\rho) = \rho^\gamme, \geq 1$. First we show that every Riemann problem whose one dimensional self-similar solution consists of two shocks admits also in_nitely many two dimensional admissible bounded weak solutions (not containing vacuum) generated by the method of De Lellis and $Sz\’{e}kelyhidi$ [11], [12]. Moreover we prove that for some of these Riemann problems and for $1\leq < 3$ such solutions have greater energy dissipation rate than the self-similar solution emanating from the same Riemann data. We therefore show that the maximal dissipation criterion proposed by Dafermos in [7] does not favour the classical self similar solutions.

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Riemann hypothesis

In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Man

Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a d

Weak solution

In mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless

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Elisabetta Chiodaroli, Camillo De Lellis

We consider the isentropic compressible Euler system in 2 space dimensions with pressure law p () = (2) and we show the existence of classical Riemann data, i.e. pure jump discontinuities across a line, for which there are infinitely many admissible bounded weak solutions (bounded away from the void). We also show that some of these Riemann data are generated by a 1-dimensional compression wave: our theorem leads therefore to Lipschitz initial data for which there are infinitely many global bounded admissible weak solutions. (c) 2015 Wiley Periodicals, Inc.

This overview is concerned with the well-posedness problem for the isentropic compressible Euler equations of gas dynamics. The results we present are in line with the programof investigatingthe efficiency of different selection criteria proposed in the literature in order to weed out non-physical solutions to more-dimensional systems of conservation laws and they build upon the method of convex integration developed by De Lellis and Sz,kelyhidi for the incompressible Euler equations. Mainly following [5], we investigate the role of the maximal dissipation criterion proposed by Dafermos in [6]: we prove how, for specific pressure laws, some non-standard (i.e. constructed via convex integration methods) solutions to the Riemann problem for the isentropic Euler system in two space dimensions have greater energy dissipation rate than the classical self-similar solution emanating from the same Riemann data. We therefore show that the maximal dissipation criterion proposed by Dafermos does not favour in general the self-similar solutions.