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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[C1]**G.-Q. Chen,*The compensated compactness method and the system of isentropic gas dynamics*, MSRI Preprint 00527-91 (1990).**[C2]**Gui Qiang Chen,*The method of quasidecoupling for discontinuous solutions to conservation laws*, Arch. Rational Mech. Anal.**121**(1992), no. 2, 131–185. MR**1188491**, 10.1007/BF00375416**[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), no. 2, 373–392. MR**0430536****[Da]**Constantine M. Dafermos,*Estimates for conservation laws with little viscosity*, SIAM J. Math. Anal.**18**(1987), no. 2, 409–421. MR**876280**, 10.1137/0518031**[Di]**R. J. DiPerna,*Convergence of approximate solutions to conservation laws*, Arch. Rational Mech. Anal.**82**(1983), no. 1, 27–70. MR**684413**, 10.1007/BF00251724**[E]**Lawrence C. Evans and Ronald F. Gariepy,*Measure theory and fine properties of functions*, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR**1158660****[Gl]**James Glimm,*Solutions in the large for nonlinear hyperbolic systems of equations*, Comm. Pure Appl. Math.**18**(1965), 697–715. MR**0194770****[La]**Peter Lax,*Shock waves and entropy*, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) Academic Press, New York, 1971, pp. 603–634. MR**0393870****[Li]**Tai Ping Liu,*Solutions in the large for the equations of nonisentropic gas dynamics*, Indiana Univ. Math. J.**26**(1977), no. 1, 147–177. MR**0435618****[M]**François Murat,*L’injection du cône positif de 𝐻⁻¹ dans 𝑊^{-1,𝑞} est compacte pour tout 𝑞<2*, J. Math. Pures Appl. (9)**60**(1981), no. 3, 309–322 (French, with English summary). MR**633007****[Ta]**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**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
35Q72,
35L65,
73B30

Retrieve articles in all journals with MSC: 35Q72, 35L65, 73B30

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1995-1270660-7

Keywords:
Thermoelasticity,
viscosity method,
compensated compactness

Article copyright:
© Copyright 1995
American Mathematical Society