Abstract
In this paper we study the zero dissipation problem for a general system of conservation laws with positive viscosity. It is shown that if the solution of the problem with zero viscosity is piecewise smooth with a finite number of noninteracting shocks satisfying the entropy condition, then there exist solutions to the corresponding system with viscosity that converge to the solutions of the system without viscosity away from shock discontinuities at a rate of order ε as the viscosity coefficient ε goes to zero. The proof uses a matched asymptotic analysis and an energy estimate related to the stability theory for viscous shock profiles.
Similar content being viewed by others
References
G. Chen, The theory of compensated compactness and the system of isentropic gas dynamics, preprint, MSRI-00527-91, Mathematical Sciences Research Institute, Berkeley.
R. J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys. 91 (1983) 1–30.
P. C. Fife, Dynamics of Internal Layers and Diffusive Interfaces, CBMS-NSF Regional Conferences Series in Applied Mathematics, #53 (1988).
J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational Mech. Anal. 95 (1986) 325–344.
J. Greenberg, Estimates for fully developed shock solutions to the equation u t − v x = 0 and v t − (σ(u))x=0, Indiana Univ. Math. J. 22 (1972/73) 989–1003.
V. Guillemin & A. Pollack, Differential Topology, Prentice-Hall, Englewood Cliffs, N.J. (1974).
S. Kawashima, Systems of Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magneto-Hydrodynamics, Doctoral Thesis, Kyoto Univ. (1983).
H. O. Kreiss, Initial boundary value problem for hyperbolic systems, Comm. Pure Appl. Math. 23 (1970) 277–295.
D. Hoff & T. P. Liu, The inviscid limit for the Navier-Stokes equations of compressible isentropic flow with shock data, Indiana Univ. Math. J. 38 (1989) 861–915.
N. Kopell & L. Howard, Bifurcations and trajectories joining critical points, Adv. Math. 18 (1975) 306–358.
Peter Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, Philadelphia (1973).
D. Li & W. C. Yu, Boundary Value Problems for Quasilinear Hyperbolic Systems, Duke Univ. Press, Durham, N.C. (1985).
T. P. Liu, Nonlinear Stability of Shock Waves for Viscous Conservation Laws, Mem. Amer. Math. Soc. 328 (1985).
J. Rauch, L 2 is a continuable initial condition for Kreiss' mixed problem, Comm. Pure Appl. Math. 25 (1972) 265–285.
J. Smoller, Shock Waves and Reaction Diffusion Equations, Springer-Verlag, New York, Berlin (1983).
Z. Xin, Zero dissipation limit to rarefaction waves for the one-dimensional Navier-Stokes equations of compressible isentropic gases (preprint).
A. Volpert, The space BV and quasilinear equations, Mat. Sb. 73 (1967) 255–302; English transl, in Math. USSR, Sb. 2 (1967) 225–267.
Author information
Authors and Affiliations
Additional information
Communicated by C. M. Dafermos
Rights and permissions
About this article
Cite this article
Goodman, J., Xin, Z. Viscous limits for piecewise smooth solutions to systems of conservation laws. Arch. Rational Mech. Anal. 121, 235–265 (1992). https://doi.org/10.1007/BF00410614
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00410614