@article {2004,
title = {On the convergence rate of vanishing viscosity approximations},
journal = {Comm. Pure Appl. Math. 57 (2004) 1075-1109},
number = {SISSA;60/2003/M},
year = {2004},
publisher = {Wiley},
abstract = {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.},
doi = {10.1002/cpa.20030},
url = {http://hdl.handle.net/1963/2915},
author = {Alberto Bressan and Tong Yang}
}
@article {2004,
title = {A sharp decay estimate for positive nonlinear waves},
journal = {SIAM J. Math. Anal. 36 (2004) 659-677},
number = {SISSA;59/2003/M},
year = {2004},
publisher = {SIAM},
abstract = {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.},
doi = {10.1137/S0036141003427774},
url = {http://hdl.handle.net/1963/2916},
author = {Alberto Bressan and Tong Yang}
}
@article {1999,
title = {L-1 stability estimates for n x n conservation laws},
journal = {Arch. Ration. Mech. Anal. 149 (1999), no. 1, 1--22},
number = {SISSA;80/98/M},
year = {1999},
publisher = {Springer},
abstract = {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 {\textquoteleft}almost decreasing\\\', i.e., $\\\\Phi(u(t),v(t))-\\\\Phi(u(s),v(s))\\\\leq\\\\break O (\\\\epsilon){\textperiodcentered}(t-s)$ 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.\\\'\\\'},
doi = {10.1007/s002050050165},
url = {http://hdl.handle.net/1963/3373},
author = {Alberto Bressan and Tai-Ping Liu and Tong Yang}
}