700 research outputs found

### 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 systÃ¨me strictement hyperbolique fournies par le schÃ©ma 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

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