40 research outputs found
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, where , and
, , . The flux function is smooth and
such that has distinct real eigenvalues for any . The problem
of existence of admissible radiative shock wave is considered, i.e. existence
of a solution of the form , such that
, and , define a shock wave
for the reduced hyperbolic system, obtained by formally putting L=0. It is
proved that, if is such that ,(where denotes the -th eigenvalue of and a
corresponding right eigenvector) and , then there exists a neighborhood of such
that for any , such that the triple
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 gains smoothness when the size of the shock 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
Energy method for multi-dimensional balance laws with non-local dissipation
AbstractIn this paper, we are concerned with a class of multi-dimensional balance laws with a non-local dissipative source which arise as simplified models for the hydrodynamics of radiating gases. At first we introduce the energy method in the setting of smooth perturbations and study the stability of constants states. Precisely, we use Fourier space analysis to quantify the energy dissipation rate and recover the optimal time-decay estimates for perturbed solutions via an interpolation inequality in Fourier space. As application, the developed energy method is used to prove stability of smooth planar waves in all dimensions n⩾2, and also to show existence and stability of time-periodic solutions in the presence of the time-periodic source. Optimal rates of convergence of solutions towards the planar waves or time-periodic states are also shown provided initially L1-perturbations
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