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

Bressan, Alberto; Goatin, Paola

Differ. Integral Equ. 13, No. 10-12, 1503-1528 (2000).

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 Oleĭnik type, conserning the decay of positive waves.

Reviewer: Georgii Sviridyuk (Chelyabinsk)

MSC:

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

Keywords:

continuous semigroup; Cauchy problem; entropy condition of Oleinik type