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Maximal Dissipation and Well-posedness for the Compressible Euler System

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Abstract

We discuss the problem of well-posedness of the compressible (barotropic) Euler system in the framework of weak solutions. The principle of maximal dissipation introduced by C.M. Dafermos is adapted and combined with the concept of admissible weak solutions. We use the method of convex integration in the spirit of the recent work of C.DeLellis and L.Székelyhidi to show various counterexamples to well-posedness. On the other hand, we conjecture that the principle of maximal dissipation should be retained as a possible criterion of uniqueness as it is violated by the oscillatory solutions obtained in the process of convex integration.

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Authors and Affiliations

  1. Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 115 67, Prague 1, Czech Republic

    Eduard Feireisl

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  1. Eduard Feireisl
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Correspondence to Eduard Feireisl.

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The research of E.F. leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC Grant Agreement 320078.

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Feireisl, E. Maximal Dissipation and Well-posedness for the Compressible Euler System. J. Math. Fluid Mech. 16, 447–461 (2014). https://doi.org/10.1007/s00021-014-0163-8

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  • DOI: https://doi.org/10.1007/s00021-014-0163-8

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