Viscous limits for piecewise smooth solutions to systems of conservation laws. (English) Zbl 0792.35115

Let \(u_ 1 + f(u)_ x\) be a strictly hyperbolic system of conservation laws (one space dimension) and \(u^ \varepsilon_ t + f(u^ \varepsilon)_ x = eu^ \varepsilon_{xx}\) the related viscous system. For scalar laws it is well known that weak entropic solutions \(u\) of the first system may be recovered as strong limits for \(\varepsilon \to 0\) of solutions \(u^ \varepsilon\) of the second; this is also true for some special \(2 \times 2\) systems. In this paper the authors prove that the same holds true for general \(n \times n\) systems, in the case that the solution \(u\) has only a finite number of noninteracting entropic shocks. More precisely they prove that \[ \sup_{0 \leq t \leq T} \int | u(x,t) - u^ \varepsilon(x,t)| dx \leq C_ \eta\varepsilon^ \eta \] for any \(\eta \in (0,1)\); outside the shock curves the error is \(O(1)\varepsilon\). The proof matches two different asymptotic expansions, one near and the other away from the shock curves; these expansions are justified through an energy estimate related to the stability theory for viscous shock profiles [T. Liu, Nonlinear stability of shock waves for viscous conservation laws. Mem. Am. Math. Soc. 328 (1985; Zbl 0617.35058)].
Reviewer: A.Corli (Ferrara)


35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
35C20 Asymptotic expansions of solutions to PDEs


Zbl 0617.35058
Full Text: DOI


[1] G. Chen, The theory of compensated compactness and the system of isentropic gas dynamics, preprint, MSRI-00527-91, Mathematical Sciences Research Institute, Berkeley.
[2] R. J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys. 91 (1983) 1-30. · Zbl 0533.76071
[3] P. C. Fife, Dynamics of Internal Layers and Diffusive Interfaces, CBMS-NSF Regional Conferences Series in Applied Mathematics, #53 (1988). · Zbl 0684.35001
[4] J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational Mech. Anal. 95 (1986) 325-344. · Zbl 0631.35058
[5] J. Greenberg, Estimates for fully developed shock solutions to the equation u t ? v x = 0 and v t ? (?(u))x=0, Indiana Univ. Math. J. 22 (1972/73) 989-1003. · Zbl 0259.35051
[6] V. Guillemin & A. Pollack, Differential Topology, Prentice-Hall, Englewood Cliffs, N.J. (1974).
[7] S. Kawashima, Systems of Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magneto-Hydrodynamics, Doctoral Thesis, Kyoto Univ. (1983).
[8] H. O. Kreiss, Initial boundary value problem for hyperbolic systems, Comm. Pure Appl. Math. 23 (1970) 277-295. · Zbl 0193.06902
[9] 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. · Zbl 0674.76047
[10] N. Kopell & L. Howard, Bifurcations and trajectories joining critical points, Adv. Math. 18 (1975) 306-358. · Zbl 0361.34026
[11] Peter Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, Philadelphia (1973). · Zbl 0268.35062
[12] D. Li & W. C. Yu, Boundary Value Problems for Quasilinear Hyperbolic Systems, Duke Univ. Press, Durham, N.C. (1985). · Zbl 0627.35001
[13] T. P. Liu, Nonlinear Stability of Shock Waves for Viscous Conservation Laws, Mem. Amer. Math. Soc. 328 (1985). · Zbl 0576.35077
[14] J. Rauch, L 2 is a continuable initial condition for Kreiss’ mixed problem, Comm. Pure Appl. Math. 25 (1972) 265-285. · Zbl 0226.35056
[15] J. Smoller, Shock Waves and Reaction Diffusion Equations, Springer-Verlag, New York, Berlin (1983). · Zbl 0508.35002
[16] Z. Xin, Zero dissipation limit to rarefaction waves for the one-dimensional Navier-Stokes equations of compressible isentropic gases (preprint). · Zbl 0804.35108
[17] A. Volpert, The space BV and quasilinear equations, Mat. Sb. 73 (1967) 255-302; English transl, in Math. USSR, Sb. 2 (1967) 225-267.
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