Preprint 1998-030

L^1 Stability Estimates for n x n Conservation Laws

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 ``almost decreasing'' i.e. $$\Phi\big( u(t),~v(t)\big)-\Phi\big( u(s),~v(s)\big)\leq \O(\ve)\cdot (t-s)\qquad\hbox{for all}~~t>s\geq 0,$$ for every couple of $\ve$-approximate solutions $u,v$ with small total variation, generated by a wave front tracking algorithm. The small parameter $\ve$ 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 a $n\times n$ system of conservation laws.

Paper:

Available as PostScript.

Title:

L^1 stability estimates for n x n conservation laws

Author(s):

Alberto Bressan, <bressan@sissa.it>

Tai-Ping Liu, <liu@gauss.Stanford.edu>

Tong Yang, <matyang@math.cityu.edu.hk>

Publishing information:

To appear in Arch. Rat. Mech. Anal.

Comments:

Revised version, October 11 1998.

Submitted by:

<wens@ifi.uio.no> August 11 1998.