IUMJ

Title: Shock waves for radiative hyperbolic-elliptic systems

Authors: Corrado Lattanzio, Corrado Mascia and Denis Serre

Issue: Volume 56 (2007), Issue 5, 2601-2640

Abstract:

The present paper deals with the following hyperbolic-elliptic coupled system, modelling dynamics of a gas in presence of radiation, \[ \begin{cases} u_{t} + f(u)_{x} + Lq_{x} = 0,\\ -q_{xx} + Rq + G \cdot u_{x} = 0, \end{cases} \qquad x \in \mathbb{R},\ t > 0, \] where $u \in \mathbb{R}^{n}$, $q \in \mathbb{R}$ and $R > 0$, $G$, $L \in \mathbb{R}^{n}$. The flux function $f : \mathbb{R}^n \to \mathbb{R}^n$ is smooth and such that $\nabla f$ has $n$ distinct real eigenvalues for any $u$.\par The problem of existence of \emph{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 $u_{\pm} \in \mathbb{R}^n$, $s \in \mathbb{R}$ define a shock wave for the reduced hyperbolic system, obtained by formally putting $L = 0$ .\par It is proved that, if $u_{-}$ is such that $\nabla \lambda_{k}(u_{-}) \cdot r_{k}(u_{-}) \neq 0$ (where $\lambda_k$ 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 $u_{+} \in \mathcal{U}$, $s \in \mathbb{R}$ 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.\par The proof is based on reducing the system case to the scalar case, hence the problem of existence for the scalar case with general strictly convex fluxes is considered, generalizing existing results for the Burgers' flux $f(u) = u^{2}/2$. 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.\par Finally, the general case of nonconvex fluxes is also treated, showing similar results of existence and regularity for the profiles.