%0 Journal Article
%J Arch. Ration. Mech. Anal. 149 (1999), no. 1, 1--22
%D 1999
%T L-1 stability estimates for n x n conservation laws
%A Alberto Bressan
%A Tai-Ping Liu
%A Tong Yang
%X Let $u_t+f(u)_x=0$ be a strictly hyperbolic $n\\\\times n$ system of conservation laws, each characteristic field being linearly degenerate or genuinely nonlinear. In this paper we explicitly define a functional $\\\\Phi=\\\\Phi(u,v)$, equivalent to the $L^1$ distance, which is `almost decreasing\\\', i.e., $\\\\Phi(u(t),v(t))-\\\\Phi(u(s),v(s))\\\\leq\\\\break O (\\\\epsilon)ยท(t-s)$ for all $t>s\\\\geq 0$, for every pair of $\\\\epsilon$-approximate solutions $u,v$ with small total variation, generated by a wave-front-tracking algorithm. The small parameter $\\\\epsilon$ here controls the errors in the wave speeds, the maximum size of rarefaction fronts and the total strength of all non-physical waves in $u$ and in $v$. From the above estimate, it follows that front-tracking approximations converge to a unique limit solution, depending Lipschitz continuously on the initial data, in the $L^1$ norm. This provides a new proof of the existence of the standard Riemann semigroup generated by an $n\\\\times n$ system of conservation laws.\\\'\\\'
%B Arch. Ration. Mech. Anal. 149 (1999), no. 1, 1--22
%I Springer
%G en_US
%U http://hdl.handle.net/1963/3373
%1 957
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-11-28T18:35:48Z\\nNo. of bitstreams: 1\\nBressan_Liu.pdf: 1602231 bytes, checksum: fe5990668e708e14f93a4f78715b929c (MD5)
%R 10.1007/s002050050165