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Transactions of the American Mathematical Society

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The vanishing viscosity method in one-dimensional thermoelasticity

Authors: Gui Qiang Chen and Constantine M. Dafermos
Journal: Trans. Amer. Math. Soc. 347 (1995), 531-541
MSC: Primary 35Q72; Secondary 35L65, 73B30
MathSciNet review: 1270660
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Abstract: The vanishing viscosity method is applied to the system of conservation laws of mass, momentum, and energy for a special class of one-dimensional thermoelastic media that do not conduct heat. Two types of vanishing "viscosity" are considered: Newtonian and artificial, in both cases accompanied by vanishing heat conductivity. It is shown that in either case one can pass to the zero viscosity limit by the method of compensated compactness, provided that velocity and pressure are uniformly bounded. Oscillations in the entropy field may propagate along the linearly degenerate characteristic field but do not affect the compactness of the velocity field or the pressure field. A priori bounds on velocity and pressure are established, albeit only for the case of artificial viscosity.

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

  • [C1] G.-Q. Chen, The compensated compactness method and the system of isentropic gas dynamics, MSRI Preprint 00527-91 (1990).
  • [C2] -, The method of quasidecoupling for discontinuous solutions to conservation laws, Arch. Rational Mech. Anal. 121 (1992), 131-185. MR 1188491 (94b:35156)
  • [CCS] K. N. Chueh, C. C. Conley, and J. A. Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. Math. J. 26 (1977), 372-411. MR 0430536 (55:3541)
  • [Da] C. M. Dafermos, Estimates for conservation laws with little viscosity, SIAM J. Math. Anal. 18 (1987), 409-421. MR 876280 (88h:35071)
  • [Di] R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1983), 27-70. MR 684413 (84k:35091)
  • [E] L. C. Evans and R. Gariepy, Measure theory and fine properties of functions, CRC Press, Ann Arbor, MI, 1992. MR 1158660 (93f:28001)
  • [Gl] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697-715. MR 0194770 (33:2976)
  • [La] P. D. Lax, Shock waves and entropy, Contributions to Functional Analysis (E. A. Zarantonello, ed.), Academic Press, New York, 1971, pp. 603-634. MR 0393870 (52:14677)
  • [Li] T.-P. Liu, Solutions in the large for the equations of nonisentropic gas dynamics, Indiana Univ. Math. J. 26 (1977), 147-177. MR 0435618 (55:8576)
  • [M] F. Murat, L'injection du cone positif de $ {H^{ - 1}}$ dans $ {W^{ - 1,q}}$ est compacte pur tout $ q < 2$, J. Math. Pures Appl. 60 (1981), 309-322. MR 633007 (83b:46045)
  • [Ta] L. Tartar, Compensated compactness and applications to partial differential equations, Pitman Res. Notes Math., vol. 39, Nonlinear Analysis and Mechanics (R. J. Knops, ed.), Pitman Press, New York, 1979. MR 584398 (81m:35014)

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Keywords: Thermoelasticity, viscosity method, compensated compactness
Article copyright: © Copyright 1995 American Mathematical Society

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