# An overview of some recent results on the Euler system of isentropic gas dynamics

- 74 Downloads

## Abstract

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 Székelyhidi 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.

## Keywords

hyperbolic systems of conservation laws Riemann problem admissible solutions entropy rate criterion ill-posedness convex integration## Mathematical subject classification

Primary: 35L65 Secondary: 35L67, 35L45## Preview

Unable to display preview. Download preview PDF.

## References

- [1]A. Bressan. Hyperbolic systems of conservation laws. The one-dimensional Cauchy problem. Oxford Lecture Series in Mathematics and its Applications, 20. Oxford University Press, Oxford (2000).zbMATHGoogle Scholar
- [2]E. Chiodaroli.
*A counterexample to well-posedeness of entropy solutions to the compressible Euler system*. J. Hyperbol. Differ. Eq.,**11**(2014), 493–519.MathSciNetCrossRefzbMATHGoogle Scholar - [3]E. Chiodaroli, C. De Lellis and O. Kreml.
*Global ill-posedness of the isentropic system of gas dynamics*. Comm. Pure Appl. Math.,**68**(2015), 1157–1190.MathSciNetCrossRefzbMATHGoogle Scholar - [4]E. Chiodaroli, E. Feireisl and O. Kreml.
*On the weak solutions to the equations of a compressible heat conducting gas*. Annales IHP-ANL,**32**(2015), 225–243.MathSciNetzbMATHGoogle Scholar - [5]E. Chiodaroli and O. Kreml.
*On the Energy Dissipation Rate of Solutions to the Compressible Isentropic Euler System*. Arch. Ration. Mech. Anal.,**214**(2014), 1019–1049.MathSciNetCrossRefzbMATHGoogle Scholar - [6]C.M. Dafermos.
*The entropy rate admissibility criterion for solutions of hyperbolic conservation laws*. J. Differential Equations,**14**(1973), 202–212.MathSciNetCrossRefzbMATHGoogle Scholar - [7]C.M. Dafermos.
*Maximal dissipation in equations of evolution*. J. Differential Equations,**252**(2012), 567–587.MathSciNetCrossRefzbMATHGoogle Scholar - [8]C.M. Dafermos. Hyperbolic conservation laws in continuum physics, vol. 325 of Grundleheren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Third edition. Springer, Berlin (2010).CrossRefzbMATHGoogle Scholar
- [9]C. De Lellis and L.J. Székelyhidi.
*The Euler equations as a differential inclusion*. Ann. Math.,**170**(3) (2009), 1417–1436.MathSciNetCrossRefzbMATHGoogle Scholar - [10]C. De Lellis and L.J. Székelyhidi.
*On admissibility criteria for weak solutions of the Euler equations*. Arch. Ration. Mech. Anal.,**195**(1) (2010), 225–260.MathSciNetCrossRefzbMATHGoogle Scholar - [11]C. De Lellis and L.J. Székelyhidi.
*The h-principle and the equations of fluid dynamics*. Bull. Amer. Math. Soc. (N.S.),**49**(3) (2012), 347–375.MathSciNetCrossRefzbMATHGoogle Scholar - [12]V. Elling.
*A possible counterexample to well posedeness of entropy solutions and to Godunov scheme convergence*. Math. Comp.,**75**(2006), 1721–1733.MathSciNetCrossRefzbMATHGoogle Scholar - [13]E. Feireisl.
*Maximal dissipation and well-posedness for the compressible Euler system*. Arch. Ration.Mech. Anal.,**16**(2014), 447–461.MathSciNetzbMATHGoogle Scholar - [14]L. Hsiao.
*The entropy rate admissibility criterion in gas dynamics*. Journal of Differential Equations,**38**(1980), 226–238.MathSciNetCrossRefzbMATHGoogle Scholar - [15]D. Serre. Systems of conservation laws. 1.CambridgeUniversity Press, Cambridge, (1999). Hyperbolicity, entropies, shock waves. Translated from the 1996 French original by I.N. Sneddon.CrossRefGoogle Scholar
- [16]L.J. Székelyhidi.
*Weak solutions to the incompressible Euler equations with vortex sheet initial data*. C. R. Acad. Sci. Paris, Ser. I,**349**(19-20) (2011), 1063–1066.MathSciNetCrossRefzbMATHGoogle Scholar