## Differential and Integral Equations

### Stability of $L^\infty$ solutions of Temple class systems

#### 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 ${{\bf L}}^\infty$ functions, with possibly unbounded variation. Trajectories depend Lipschitz continuously on the initial data, in the ${{\bf L}}^1$ 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 Oleinik type, concerning the decay of positive waves.

#### Article information

Source
Differential Integral Equations, Volume 13, Number 10-12 (2000), 1503-1528.

Dates
First available in Project Euclid: 21 December 2012

https://projecteuclid.org/euclid.die/1356061137

Mathematical Reviews number (MathSciNet)
MR1787079

Zentralblatt MATH identifier
1047.35095

Subjects
Primary: 35L65: Conservation laws
Secondary: 35B35: Stability

#### Citation

Bressan, Alberto; Goatin, Paola. Stability of $L^\infty$ solutions of Temple class systems. Differential Integral Equations 13 (2000), no. 10-12, 1503--1528. https://projecteuclid.org/euclid.die/1356061137