RELATIVE ENTROPY AND COMPRESSIBLE POTENTIAL FLOW

doi:10.1016/S0252-9602(15)30020-5

Abstract

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

CLC Number:

35L67

Volker ELLING. RELATIVE ENTROPY AND COMPRESSIBLE POTENTIAL FLOW[J].Acta mathematica scientia,Series B, 2015, 35(4): 763-776.

Trendmd

References

[1] Bressan A, Crasta G, Piccoli B. Well-posedness of the cauchy problem for n × n systems of conservation laws. Memoirs Amer Math Soc, 2000, (694)
[2] Bressan A, LeFloch P. Uniqueness of weak solutions to systems of conservation laws. Arch Rat Mech Anal, 1997, 140: 301-317
[3] Chen Gui-Qiang, Dafermos C M, Slemrod M, Wang Dehua. On two-dimensional sonic-subsonic flow. Comm Math Phys, 2007, 271(3): 635-647
[4] Chipman R, Jameson A. Fully conservative numerical solutions for unsteady irrotational transonic flow about airfoils. Technical Report, 1979
[5] Dafermos C. The second law of thermodynamics and stability. Arch Rat Mech Anal, 1979, 70: 167-179
[6] DiPerna R J. Uniqueness of solutions to hyperbolic conservation laws. Indiana Univ Math J, 1979, 28: 137-188
[7] Elling V. Existence of algebraic vortex spirals//Proceedings of the 13th Conference on Hyperbolic Problems (HYP2010)
[8] Elling V. A possible counterexample to well-posedness of entropy solutions and to Godunov scheme con- vergence. Math Comp, 2006, 75: 1721-1733
[9] Elling V. The carbuncle phenomenon is incurable. Acta Math Sci, 2009, 29B(6): 1647-1656
[10] Elling V. Regular reflection in self-similar potential flow and the sonic criterion. Commun Math Anal, 2010, 8(2): 22-69
[11] Elling V, Liu Tai-Ping. Supersonic flow onto a solid wedge. Comm Pure Appl Math, 2008, 61(10): 1347- 1448
[12] Friedrichs K O, Lax P. Systems of conservation laws with a convex extension. Proc Nat Acad Sci USA, 1971, 68(8): 1686-1688
[13] Glimm J. Solutions in the large for nonlinear hyperbolic systems of equations. Comm Pure Appl Math, 1965, 18: 697-715
[14] John F. Formation of singularities in one-dimensional nonlinear wave propagation. Comm Pure Appl Math, 1974, 27: 377-405
[15] Lax P D. Hyperbolic systems of conservation laws II. Comm Pure Appl Math, 1957, 10: 537-566
[16] Lax P D. Development of singularities of solutions of nonlinear hyperbolic partial differential equations. J Math Phys, 1964, 5: 611-613
[17] Lellis C De, Székelyhidi Jr L. On admissibility criteria for weak solutions of the Euler equations. Arch Rat Mech Anal, 2010, 195(1): 225-260
[18] Liu Tai-Ping. The Riemann problem for general 2×2 conservation laws. Trans Amer Math Soc, 1974, 199: 89-112
[19] Liu Tai-Ping. The deterministic version of the Glimm scheme. Comm Math Phys, 1977, 57: 135-148
[20] Liu Tai-Ping. Development of singularities in the nonlinear waves for quasi-linear hyperbolic partial differ- ential equations. J Diff Eqs, 1979, 33: 92-111
[21] Liu Tai-Ping. Admissible solutions of hyperbolic conservation laws. Memoirs Amer Math Soc, 1981, (240)
[22] Liu Tai-Ping, Yang Tong. L₁ stability for 2×2 systems of hyperbolic conservation laws. J Amer Math Soc, 1999, 12: 729-774
[23] Liu Tai-Ping, Yang Tong. Well-posedness theory for hyperbolic conservation laws. Comm Pure Appl Math, 1999, 52: 1553-1586
[24] Morawetz C S. On steady transonic flow by compensated compactness. Methods Appl Anal, 1995, 2: 257-268
[25] Osher S, Hafez M, Whitlow W. Entropy condition satisfying approximations for the full potential equation of transonic flow. Math Comp, 1985, 44: 1-29
[26] Sideris T. Formation of singularities in three-dimensional compressible fluids. Comm Math Phys, 1985, 101: 475-485