Viscous Limits for strong shocks of one-dimensional systems of conservation laws

Rousset, Frédéric

Journées équations aux dérivées partielles (2002), article no. 16, 11 p.

On considère un système hyperbolique de lois de conservation monodimensionnel $u_t+f{\left(u\right)}_x=0$, et une solution continue par morceaux avec un seul choc de ce système. En supposant qu’en tout point de discontinuité, il existe un profil visqueux linéairement stable, on montre qu’il existe une solution du système avec viscosité $u_t+f{\left(u\right)}_x=ϵu_{xx}$ qui tend vers la solution discontinue dans $L^{\infty }\left([0,T]L^1\right)$ lorsque la viscosité tend vers zéro.

We consider a piecewise smooth solution of a one-dimensional hyperbolic system of conservation laws with a single noncharacteristic Lax shock. We show that it is a zero dissipation limit assuming that there exist linearly stable viscous profiles associated with the discontinuities. In particular, following the approach of Grenier and Rousset (2001), we replace the smallness condition obtained by energy methods in Goodman and Xin (1992) by a weaker spectral assumption.

[1] S. Bianchini and A. Bressan. Vanishing viscosity solutions of nonlinear hyperbolic systems. Preprint, 2001.

[2] B. Desjardins and E. Grenier. Linear instability implies nonlinear instability for various boundary layers. Preprint ENS-Lyon.

[3] R. J. Diperna. Convergence of the viscosity method for isentropic gas dynamics. Comm. Math. Phys., 91(1):1-30, 1983. | MR | Zbl

[4] R. A. Gardner and K. Zumbrun. The gap lemma and geometric criteria for instability of viscous shock profiles. Comm. Pure Appl. Math., 51(7):797-855, 1998. | MR | Zbl

[5] J. Goodman. Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch. Rational Mech. Anal., 95(4):325-344, 1986. | MR | Zbl

[6] J. Goodman and Z. P. Xin. Viscous limits for piecewise smooth solutions to systems of conservation laws. Arch. Rational Mech. Anal., 121(3):235-265, 1992. | MR | Zbl

[7] E. Grenier and F. Rousset. Stability of one-dimensional boundary layers by usingGreen's functions. Comm. Pure Appl. Math., 54(11):1343-1385, 2001. | MR | Zbl

[8] O. Guès, G. Métiver, M. Williams and K. Zumbrun. Multidimensional viscous shocks II : the small viscosity limit. Preprint, 2002

[9] O. Guès, M. Williams. Curved shocks as viscous limits : a boundary problem approach. Preprint, 2001 | MR

[10] G. Kreiss and H.-O. Kreiss. Stability of systems of viscous conservation laws. Comm. Pure Appl. Math., 51(11-12):1397-1424, 1998. | MR | Zbl

[11] T.-P. Liu. Pointwise convergence to shock waves for viscous conservation laws. Comm. Pure Appl. Math., 50(11):1113-1182, 1997. | MR | Zbl

[12] G. MŽÉtivier and K. Zumbrun. Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems. Preprint, 2002

[13] K. J. Palmer. Exponential dichotomies and transversal homoclinic points. J. Differential Equations, 55(2):225-256, 1984. | MR | Zbl

[14] F. Rousset. Viscous limits for strong shocks of systems of conservation laws. Preprint, 2001 XVI-11

[15] A. I. Volpert. Spaces BV and quasilinear equations. Mat. Sb.(N.S.), 73(115):255-302, 1967. | MR | Zbl

[16] S. H. Yu. Zero-dissipation limit of solutions with shocks for systems of hyperbolic conservation laws. Arch. Ration. Mech. Anal., 146(4):275-370, 1999. | MR | Zbl

[17] K. Zumbrun and P. Howard. Pointwise semigroup methods and stability of viscous shock waves. Indiana Univ. Math. J., 47(3):741-871, 1998. | MR | Zbl

[18] K. Zumbrun and D. Serre. Viscous and inviscid stability of multidimensional planar shock fronts. Indiana Univ. Math. J., 48(3):937-992, 1999 | MR | Zbl