## Weak solutions with a shock to a model system of the radiating gas.(English)Zbl 0915.76074

The paper deals with a reduced model for polytropic gas with radiative heat flow. This model consists of a Burgers type equation coupled with an elliptic linear equation: $$u_t + u u_x + q_x = 0$$, $$-q_{xx} + q + u_x = 0$$, for which the Riemann problem is investigated. It is shown that for $$u_->u_+$$, there exists a unique weak solution according to a specified entropy criterion. Moreover, if $$| u_--u_+| \leq\frac{1}{2}$$, then the solution is global and converges to a travelling wave with decay rate $$t^{-1/4}$$. Travelling waves are studied in a companion paper that has appeared in SIAM J. Math. Anal. 30, No. 1, 95-117 (1999).
Reviewer: S.Benzoni (Lyon)

### MSC:

 76N15 Gas dynamics (general theory) 76L05 Shock waves and blast waves in fluid mechanics 80A20 Heat and mass transfer, heat flow (MSC2010)