L1 stability estimates for n n conservation laws

Alberto Bressan Tai-Ping Liu Tong Yang

Analysis of PDEs mathscidoc:1912.43925

Archive for rational mechanics and analysis, 149, (1), 1-22, 1999.10
. 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., $$ \Phi\big( u(t),~v(t)\big)-\Phi\big( u(s),~v(s)\big)\leq \O(\ve)\cdot (t-s)\quad\hbox{for all}~~t&gt;s\geq 0,$$ for every pair of <i></i>-approximate solutions <i>u</i>, <i>v</i> with small total variation, generated by a wave front tracking algorithm. The small parameter <i></i> here controls the errors in the wave speeds, the maximum size of rarefaction fronts and the total strength of all non-physical waves in <i>u</i> and in <i>v</i>. 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 ${\vec L}^1$ norm. This provides a new proof of the existence of the
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  title={L1 stability estimates for n n conservation laws},
  author={Alberto Bressan, Tai-Ping Liu, and Tong Yang},
  booktitle={Archive for rational mechanics and analysis},
Alberto Bressan, Tai-Ping Liu, and Tong Yang. L1 stability estimates for n n conservation laws. 1999. Vol. 149. In Archive for rational mechanics and analysis. pp.1-22. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210711448085489.
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