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$$L^1$$ stability estimates for $$n\times n$$ conservation laws. (English) Zbl 0938.35093
The authors give a new concise proof of the $$L^1$$-stability of solutions to the Cauchy problem for a strictly hyperbolic $$n\times n$$ system of conservation laws $${u_t+f(u)_x=0}$$. It is assumed that each characteristic field is either linearly degenerate or genuinely nonlinear. This requirement allows to apply a wave front tracking method to construct $$\varepsilon$$-approximate solutions. The authors explicitly define a functional $$\Phi=\Phi(u,v)$$, equivalent to the $$L^1$$ distance, such that $\Phi(u(t),v(t))-\Phi(u(s),v(s))=O(\varepsilon)(t-s) \quad \forall t>s\geq 0$ for every pair of $$\varepsilon$$-approximate solutions $$u,v$$ having small total variation. The above estimate implies convergence of wave front tracking approximations to an unique limit solution depending Lipschitz continuously on the initial data in the $$L^1$$-norm and hence existence of the standard Riemann semigroup generated by the system under consideration.

##### MSC:
 35L65 Hyperbolic conservation laws 35B35 Stability in context of PDEs 35L45 Initial value problems for first-order hyperbolic systems
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