数学物理学报(英文版) ›› 2015, Vol. 35 ›› Issue (4): 763-776.doi: 10.1016/S0252-9602(15)30020-5

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RELATIVE ENTROPY AND COMPRESSIBLE POTENTIAL FLOW

Volker ELLING   

  • 收稿日期:2014-11-08 出版日期:2015-07-01 发布日期:2015-07-01
  • 基金资助:

    This material is based upon work partially supported by the National Science Foundation under Grant No. NSF DMS-1054115 and by a Sloan Foundation Research Fellowship.

RELATIVE ENTROPY AND COMPRESSIBLE POTENTIAL FLOW

Volker ELLING   

  1. Department of Mathematics, East Hall, University of Michigan, 530 Church St, Ann Arbor, MI 48109, USA
  • Received:2014-11-08 Online:2015-07-01 Published:2015-07-01
  • Supported by:

    This material is based upon work partially supported by the National Science Foundation under Grant No. NSF DMS-1054115 and by a Sloan Foundation Research Fellowship.

摘要:

Compressible (full) potential flow is expressed as an equivalent first-order system of conservation laws for density and velocity v. Energy E is shown to be the only nontrivial entropy for that system in multiple space dimensions, and it is strictly convex in ρ, v if and only if |v| < c. For motivation some simple variations on the relative entropy theme of Dafer- mos/DiPerna are given, for example that smooth regions of weak entropy solutions shrink at finite speed, and that smooth solutions force solutions of singular entropy-compatible per- turbations to converge to them. We conjecture that entropy weak solutions of compressible potential flow are unique, in contrast to the known counterexamples for the Euler equations.

关键词: potential flow, entropy, relative entropy, shock

Abstract:

Compressible (full) potential flow is expressed as an equivalent first-order system of conservation laws for density and velocity v. Energy E is shown to be the only nontrivial entropy for that system in multiple space dimensions, and it is strictly convex in ρ, v if and only if |v| < c. For motivation some simple variations on the relative entropy theme of Dafer- mos/DiPerna are given, for example that smooth regions of weak entropy solutions shrink at finite speed, and that smooth solutions force solutions of singular entropy-compatible per- turbations to converge to them. We conjecture that entropy weak solutions of compressible potential flow are unique, in contrast to the known counterexamples for the Euler equations.

Key words: potential flow, entropy, relative entropy, shock

中图分类号: 

  • 35L67