## Differential and Integral Equations

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

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

Alberto Bressan and Paola Goatin

#### 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

**Permanent link to this document**

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