Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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


Author: Gui-Qiang Chen
Journal: Proc. Amer. Math. Soc. 125 (1997), 2981-2986
MSC (1991): Primary 35K55, 35L65; Secondary 76N15, 35L60, 65M06
DOI: https://doi.org/10.1090/S0002-9939-97-03946-4
MathSciNet review: 1403118
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 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 https://doi.org/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.
  • Lax, P., Weak solutions of nonlinear hyperbolic equations and their numerical computation, Comm. Pure Appl. Math. 7 (1954), 159-193.
  • 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
  • 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
  • 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 https://doi.org/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.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 35K55, 35L65, 76N15, 35L60, 65M06

Retrieve articles in all journals with MSC (1991): 35K55, 35L65, 76N15, 35L60, 65M06


Additional Information

Gui-Qiang Chen
Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208-2730
MR Author ID: 249262
ORCID: 0000-0001-5146-3839
Email: gqchen@math.nwu.edu

Keywords: DiPerna’s lemma, remarks, gap, fix, bypass, viscosity method, finite difference methods, isentropic Euler equations
Received by editor(s): May 16, 1996
Communicated by: James Glimm
Article copyright: © Copyright 1997 American Mathematical Society