Stability of \(L^ \infty\) solutions of Temple class systems. (English) Zbl 1047.35095

Authors’ abstract: Let \(u_t + f(u)_x=0\) be a strictly hyperbolic, genuinely nonlinear system of conservation laws of Temple class. In this paper, a continuous semigroup of solutions is constructed on a domain of \(L^\infty\) functions, with possibly unbounded variation. Trajectories depend Lipschitz continuously on the initial data, in the \(L^1\) distance. Moreover, we show that a weak solution of the Cauchy problem coincides with the corresponding semigroup trajectory if and only if it satisfies an entropy condition of Oleń≠nik type, conserning the decay of positive waves.


35L65 Hyperbolic conservation laws
47D06 One-parameter semigroups and linear evolution equations