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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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