%0 Journal Article
%J Comm. Pure Appl. Math. 57 (2004) 1075-1109
%D 2004
%T On the convergence rate of vanishing viscosity approximations
%A Alberto Bressan
%A Tong Yang
%X Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound $\\\\big\\\\|u(t,\\\\cdot)-u^\\\\ve(t,\\\\cdot)\\\\big\\\\|_{\\\\L^1}= \\\\O(1)(1+t)\\\\cdot \\\\sqrt\\\\ve|\\\\ln\\\\ve|$ on the distance between an exact BV solution $u$ and a viscous approximation $u^\\\\ve$, letting the viscosity coefficient $\\\\ve\\\\to 0$. In the proof, starting from $u$ we construct an approximation of the viscous solution $u^\\\\ve$ by taking a mollification $u*\\\\phi_{\\\\strut \\\\sqrt\\\\ve}$ and inserting viscous shock profiles at the locations of finitely many large shocks, for each fixed $\\\\ve$. Error estimates are then obtained by introducing new Lyapunov functionals which control shock interactions, interactions between waves of different families and by using sharp decay estimates for positive nonlinear waves.
%B Comm. Pure Appl. Math. 57 (2004) 1075-1109
%I Wiley
%G en_US
%U http://hdl.handle.net/1963/2915
%1 1785
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-09-11T13:30:25Z\\nNo. of bitstreams: 1\\nmath.AP0307141.pdf: 265243 bytes, checksum: 795adebd067228364ac1240add5f7b02 (MD5)
%R 10.1002/cpa.20030
%0 Journal Article
%J SIAM J. Math. Anal. 36 (2004) 659-677
%D 2004
%T A sharp decay estimate for positive nonlinear waves
%A Alberto Bressan
%A Tong Yang
%X We consider a strictly hyperbolic, genuinely nonlinear system of conservation laws in one space dimension. A sharp decay estimate is proved for the positive waves in an entropy weak solution. The result is stated in terms of a partial ordering among positive measures, using symmetric rearrangements and a comparison with a solution of Burgers\\\' equation with impulsive sources.
%B SIAM J. Math. Anal. 36 (2004) 659-677
%I SIAM
%G en_US
%U http://hdl.handle.net/1963/2916
%1 1784
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-09-11T13:37:38Z\\nNo. of bitstreams: 1\\nmath.AP0307140.pdf: 167236 bytes, checksum: 9e7a6fecd3de67843ca5c73f13bd841d (MD5)
%R 10.1137/S0036141003427774
%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