## Advances in Differential Equations

- Adv. Differential Equations
- Volume 2, Number 4 (1997), 643-666.

### Stiff well-posedness for hyperbolic systems with large relaxation terms (linear constant-coefficient problems)

Jens Lorenz and H. Joachim Schroll

#### Abstract

The Cauchy problem for linear constant--coefficient hyperbolic systems $u_t+Au_x=(1/\delta)Bu$ is analyzed. Here $(1/\delta)Bu$ is a large relaxation term, and we are mostly interested in the critical case where $B$ has a nontrivial nullspace. A concept of stiff well--posedness is introduced that ensures solution estimates independent of $0<\delta \leq 1$. Under suitable assumptions, we prove convergence of the $L_2$--solution to a limit as $\delta$ tends to zero. The limit solves a reduced strongly hyperbolic system without zero--order term, the so--called equilibrium system, and we present a method to determine this limit system. For 2$\times$2 systems the requirement of stiff well--posedness is shown to be equivalent to the well--known subcharacteristic condition, but in general the subcharacteristic condition is not sufficient for stiff well--posedness. The theory is illustrated by examples.

#### Article information

**Source**

Adv. Differential Equations, Volume 2, Number 4 (1997), 643-666.

**Dates**

First available in Project Euclid: 23 April 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1366741152

**Mathematical Reviews number (MathSciNet)**

MR1441860

**Zentralblatt MATH identifier**

1023.35519

**Subjects**

Primary: 35L45: Initial value problems for first-order hyperbolic systems

Secondary: 35E20: General theory 47N20: Applications to differential and integral equations 65M99: None of the above, but in this section

#### Citation

Lorenz, Jens; Schroll, H. Joachim. Stiff well-posedness for hyperbolic systems with large relaxation terms (linear constant-coefficient problems). Adv. Differential Equations 2 (1997), no. 4, 643--666. https://projecteuclid.org/euclid.ade/1366741152