Shock waves for radiative hyperbolic--elliptic systems

Abstract

The present paper deals with the following hyperbolic--elliptic coupled system, modelling dynamics of a gas in presence of radiation, $u_{t}+ f(u)_{x} +Lq_{x}=0, -q_{xx} + Rq +G\cdot u_{x}=0,$ where $u\in\R^{n}$, $q\in\R$ and $R>0$, $G$, $L\in\R^{n}$. The flux function $f : \R^n\to\R^n$ is smooth and such that $\nabla f$ has $n$ distinct real eigenvalues for any $u$. The problem of existence of admissible radiative shock wave is considered, i.e. existence of a solution of the form $(u,q)(x,t):=(U,Q)(x-st)$, such that $(U,Q)(\pm\infty)=(u_\pm,0)$, and $u_\pm\in\R^n$, $s\in\R$ define a shock wave for the reduced hyperbolic system, obtained by formally putting L=0. It is proved that, if $u_-$ is such that $\nablaλ_{k}(u_-)\cdot r_{k}(u_-)\neq 0$,(where $λ_k$ denotes the $k$-th eigenvalue of $\nabla f$ and $r_k$ a corresponding right eigenvector) and $(\ell_{k}(u_{-})\cdot L) (G\cdot r_{k}(u_{-})) >0$, then there exists a neighborhood $\mathcal U$ of $u_-$ such that for any $u_+\in{\mathcal U}$, $s\in\R$ such that the triple $(u_{-},u_{+};s)$ defines a shock wave for the reduced hyperbolic system, there exists a (unique up to shift) admissible radiative shock wave for the complete hyperbolic--elliptic system. Additionally, we are able to prove that the profile $(U,Q)$ gains smoothness when the size of the shock $|u_+-u_-|$ is small enough, as previously proved for the Burgers' flux case. Finally, the general case of nonconvex fluxes is also treated, showing similar results of existence and regularity for the profiles.