Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Cauchy problem for hyperbolic conservation laws with a relaxation term

Christian Klingenberga1 and Yun-guang Lua2

a1 Applied Mathematics Department, Heidelberg University, Heidelberg, Germany

a2 Young Scientist Lab of Mathematical Physics, Wuhan Institute of Mathematical Sciences, Academia Sinica, Wuhan, China

Abstract

This paper considers the Cauchy problem for hyperbolic conservation laws arising in chromatography:

S0308210500023088_eqnU1

with bounded measurable initial data, where the relaxation term g(δ, u, v) converges to zero as the parameter δ > 0 tends to zero. We show that a solution of the equilibrium equation

S0308210500023088_eqnU2

is given by the limit of the solutions of the viscous approximation

S0308210500023088_eqnU3

of the original system as the dissipation ε and the relaxation δ go to zero related by δ = O(ε). The proof of convergence is obtained by a simplified method of compensated compactness [2], avoiding Young measures by using the weak continuity theorem (3.3) of two by two determinants.

(Received September 12 1994)