TY - JOUR
T1 - On the convergence rate of vanishing viscosity approximations
JF - Comm. Pure Appl. Math. 57 (2004) 1075-1109
Y1 - 2004
A1 - Alberto Bressan
A1 - Tong Yang
AB - 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.
PB - Wiley
UR - http://hdl.handle.net/1963/2915
U1 - 1785
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - A sharp decay estimate for positive nonlinear waves
JF - SIAM J. Math. Anal. 36 (2004) 659-677
Y1 - 2004
A1 - Alberto Bressan
A1 - Tong Yang
AB - 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.
PB - SIAM
UR - http://hdl.handle.net/1963/2916
U1 - 1784
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - L-1 stability estimates for n x n conservation laws
JF - Arch. Ration. Mech. Anal. 149 (1999), no. 1, 1--22
Y1 - 1999
A1 - Alberto Bressan
A1 - Tai-Ping Liu
A1 - Tong Yang
AB - 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.\\\'\\\'
PB - Springer
UR - http://hdl.handle.net/1963/3373
U1 - 957
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -