Stability of scalar radiative shock profiles

This work establishes nonlinear orbital asymptotic stability of scalar radiative shock profiles, namely, traveling wave solutions to the simplified model system of radiating gas \cite{Hm}, consisting of a scalar conservation law coupled with an elliptic equation for the radiation flux. The method is based on the derivation of pointwise Green function bounds and description of the linearized solution operator. A new feature in the present analysis is the construction of the resolvent kernel for the case of an eigenvalue system of equations of degenerate type. Nonlinear stability then follows in standard fashion by linear estimates derived from these pointwise bounds, combined with nonlinear-damping type energy estimates

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

Discontinuities in numerical radiative transfer

Observations and magnetohydrodynamic simulations of solar and stellar atmospheres reveal an intermittent behavior or steep gradients in physical parameters, such as magnetic field, temperature, and bulk velocities. The numerical solution of the stationary radiative transfer equation is particularly challenging in such situations, because standard numerical methods may perform very inefficiently in the absence of local smoothness. However, a rigorous investigation of the numerical treatment of the radiative transfer equation in discontinuous media is still lacking. The aim of this work is to expose the limitations of standard convergence analyses for this problem and to identify the relevant issues. Moreover, specific numerical tests are performed. These show that discontinuities in the atmospheric physical parameters effectively induce first-order discontinuities in the radiative transfer equation, reducing the accuracy of the solution and thwarting high-order convergence. In addition, a survey of the existing numerical schemes for discontinuous ordinary differential systems and interpolation techniques for discontinuous discrete data is given, evaluating their applicability to the radiative transfer problem

Buoyancy Instabilities in Galaxy Clusters: Convection Due to Adiabatic Cosmic Rays and Anisotropic Thermal Conduction

Using a linear stability analysis and two and three-dimensional nonlinear simulations, we study the physics of buoyancy instabilities in a combined thermal and relativistic (cosmic ray) plasma, motivated by the application to clusters of galaxies. We argue that cosmic ray diffusion is likely to be slow compared to the buoyancy time on large length scales, so that cosmic rays are effectively adiabatic. If the cosmic ray pressure $p_{cr}$ is $\gtrsim 25$ of the thermal pressure, and the cosmic ray entropy ($p_{\rm cr}/\rho^{4/3}$; $\rho$ is the thermal plasma density) decreases outwards, cosmic rays drive an adiabatic convective instability analogous to Schwarzschild convection in stars. Global simulations of galaxy cluster cores show that this instability saturates by reducing the cosmic ray entropy gradient and driving efficient convection and turbulent mixing. At larger radii in cluster cores, the thermal plasma is unstable to the heat flux-driven buoyancy instability (HBI), a convective instability generated by anisotropic thermal conduction and a background conductive heat flux. Cosmic-ray driven convection and the HBI may contribute to redistributing metals produced by Type 1a supernovae in clusters. Our calculations demonstrate that adiabatic simulations of galaxy clusters can artificially suppress the mixing of thermal and relativistic plasma; anisotropic thermal conduction allows more efficient mixing, which may contribute to cosmic rays being distributed throughout the cluster volume.Comment: submitted to ApJ; 15 pages and 12 figures; abstract shortened to < 24 lines; for high resolution movies see http://astro.berkeley.edu/~psharma/clustermovie.htm

Shock Profiles for Non Equilibrium Radiating Gases

We study a model of radiating gases that describes the interaction of an inviscid gas with photons. We show the existence of smooth traveling waves called 'shock profiles', when the strength of the shock is small. Moreover, we prove that the regularity of the traveling wave increases when the strength of the shock tends to zero

Small, medium and large shock waves for non-equilibrium radiation hydrodynamic

We examine the existence of shock profiles for a hyperbolic-elliptic system arising in radiation hydrodynamics. The algebraic-differential system for the wave profile is reduced to a standard two-dimensional form that is analyzed in details showing the existence of heteroclinic connection between the two singular points of the system for any distance between the corresponding asymptotic states of the original model. Depending on the location of these asymptotic states, the profile can be either continuous or possesses at most one point of discontinuity. Moreover, a sharp threshold relative to presence of an internal absolute maximum in the temperature profile --also called {\sf Zel'dovich spike}-- is rigourously derived.Comment: 22 pages, 3 figure