Expansion of a compressible gas in vacuum

Tai-Ping Liu \cite{Liu\_JJ} introduced the notion of "physical solution' of the isentropic Euler system when the gas is surrounded by vacuum. This notion can be interpreted by saying that the front is driven by a force resulting from a H\"older singularity of the sound speed. We address the question of when this acceleration appears or when the front just move at constant velocity. We know from \cite{Gra,SerAIF} that smooth isentropic flows with a non-accelerated front exist globally in time, for suitable initial data. In even space dimension, these solutions may persist for all $t\in\R$ ; we say that they are {\em eternal}. We derive a sufficient condition in terms of the initial data, under which the boundary singularity must appear. As a consequence, we show that, in contrast to the even-dimensional case, eternal flows with a non-accelerated front don't exist in odd space dimension. In one space dimension, we give a refined definition of physical solutions. We show that for a shock-free flow, their asymptotics as both ends $t\rightarrow\pm\infty$ are intimately related to each other

Shock waves for radiative hyperbolic--elliptic systems

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 $\nabla\lambda_{k}(u_-)\cdot r_{k}(u_-)\neq 0$,(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 $u_+\in{\mathcal U}$, $s\in\R$ 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.Comment: 32 page

Conditions aux limites pour un système strictement hyperbolique fournies par le schéma de Godunov

International audienceWe study hyperbolic systems of conservation laws in one space variable, in particular the behaviour of the boundary conditions for the Godunov scheme as the space step tends to zero. Thanks to entropy estimates, we prove the convergence of the solution of the scheme towards the solution of a hyperbolic initial boundary value problem