Remarks on DiPerna’s paper “Convergence of the viscosity method for isentropic gas dynamics”

Chen, Gui-Qiang

doi:10.1090/S0002-9939-97-03946-4

Remarks on DiPerna’s paper “Convergence of the viscosity method for isentropic gas dynamics”
HTML articles powered by AMS MathViewer

by Gui-Qiang Chen PDF

Proc. Amer. Math. Soc. 125 (1997), 2981-2986 Request permission

Abstract:

Concerns have been voiced about the correctness of certain technical points in DiPerna’s paper (Comm. Math. Phys. 91 (1983), 1–30) related to the vacuum state. In this note, we provide clarifications. Our conclusion is that these concerns mainly arise from the statement of a lemma for constructing the viscous approximate solutions and some typos; however, the gap can be either fixed by correcting the statement of the lemma and the typos or bypassed by employing the finite difference methods.

References

Ronald J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys. 91 (1983), no. 1, 1–30. MR 719807
Evan K. Westwood, Complex ray methods for acoustic interaction at a fluid-fluid interface, J. Acoust. Soc. Amer. 85 (1989), no. 5, 1872–1884. MR 995741, DOI 10.1121/1.397894
Chen, G.-Q., Remarks on spherically symmetric solutions to the compressible Euler equations, Proc. Roy. Soc. Edinburgh 120A (1997).
Chen, G.-Q., Compactness Methods and Entropy Analysis for Conservation Laws, Lecture Notes, In preparation.
Xia Xi Ding, Gui Qiang Chen, and Pei Zhu Luo, Convergence of the Lax-Friedrichs scheme for the system of equations of isentropic gas dynamics. I, Acta Math. Sci. (Chinese) 7 (1987), no. 4, 467–480 (Chinese). MR 943646
Xia Xi Ding, Gui Qiang Chen, and Pei Zhu Luo, Convergence of the fractional step Lax-Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics, Comm. Math. Phys. 121 (1989), no. 1, 63–84. MR 985615
Chen, G.-Q. and LeFloch, P. G., Isentropic Euler Equations with general pressure law, Preprint, May 1997.
Chen, G.-Q. and Glimm, J., Global solutions to the compressible Euler equations with geometrical structure, Commun. Math. Phys. 180 (1996), 153–193.
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
S. K. Godunov, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Mat. Sb. (N.S.) 47 (89) (1959), 271–306 (Russian). MR 0119433
James Glimm and Andrew J. Majda (eds.), Multidimensional hyperbolic problems and computations, The IMA Volumes in Mathematics and its Applications, vol. 29, Springer-Verlag, New York, 1991. Papers from the IMA Workshop held in Minneapolis, Minnesota, April 3–14, 1989. MR 1087068, DOI 10.1007/978-1-4613-9121-0
Lawrence C. Evans, Weak convergence methods for nonlinear partial differential equations, CBMS Regional Conference Series in Mathematics, vol. 74, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. MR 1034481, DOI 10.1090/cbms/074
Frid Neto, H., Compensated Compactness Applied to Conservation Laws, Col’oquio 19, Brasileiro de Matemática, IMPA, Estrada Dona Castorina, 110-J. Botânico, CEP:22460.320, Rio de Janeiro-RJ.
Peter Constantin and Itamar Procaccia, The geometry of turbulent advection: sharp estimates for the dimensions of level sets, Nonlinearity 7 (1994), no. 3, 1045–1054. MR 1275539
Lions, P. L., Perthame, B., and Souganidis, E., Existence of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Preprint, 1995.
Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR 688146
Tung Chang and Ling Hsiao, The Riemann problem and interaction of waves in gas dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 41, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. MR 994414
John M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1976/77), no. 4, 337–403. MR 475169, DOI 10.1007/BF00279992
Denis Serre, La compacité par compensation pour les systèmes hyperboliques non linéaires de deux équations à une dimension d’espace, J. Math. Pures Appl. (9) 65 (1986), no. 4, 423–468 (French). MR 881690
L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass.-London, 1979, pp. 136–212. MR 584398
Murat, F., A survey on compensated compactness, In: Contributions to Modern Calculus of Variations, Pitman Research Notes in Math. 148 (1987), 145-183.