01240nas a2200121 4500008004300000245006600043210005900109260001000168520086800178100002101046700001501067856003601082 2004 en_Ud 00aOn the convergence rate of vanishing viscosity approximations0 aconvergence rate of vanishing viscosity approximations bWiley3 aGiven 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.1 aBressan, Alberto1 aYang, Tong uhttp://hdl.handle.net/1963/291500741nas a2200121 4500008004300000245005600043210005400099260000900153520038500162100002100547700001500568856003600583 2004 en_Ud 00aA sharp decay estimate for positive nonlinear waves0 asharp decay estimate for positive nonlinear waves bSIAM3 aWe 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.1 aBressan, Alberto1 aYang, Tong uhttp://hdl.handle.net/1963/291601453nas a2200133 4500008004300000245005600043210005500099260001300154520106200167100002101229700001801250700001501268856003601283 1999 en_Ud 00aL-1 stability estimates for n x n conservation laws0 aL1 stability estimates for n x n conservation laws bSpringer3 aLet $u_t+f(u)_x=0$ 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.\\\'\\\'1 aBressan, Alberto1 aLiu, Tai-Ping1 aYang, Tong uhttp://hdl.handle.net/1963/3373