ASYMPTOTIC STABILITY OF RAREFACTION WAVE FOR HYPERBOLIC-ELLIPTIC COUPLED SYSTEM IN RADIATING GAS

In this article, authors study the Cauch problem for a model of hyperbolic-elliptic coupled system derived from the one-dimensional system of the rudiating gas. By considering the initial data as a small disturbances of rarefaction wave of inviscid Burgers equation, the global existence of the solution to the corresponding Cauchy problem and asymptotic stability of rarefaction wave is proved. The analysis is based on a priori estimates and L^2-energy method.

Numerical Approximation of Oscillatory Solutions of Hyperbolic-Elliptic Systems of Conservation Laws by Multiresolution Schemes

The generic structure of solutions of initial value problems of hyperbolic-elliptic systems,also called mixed systems,of conservation laws is not yet fully understood.One reason for the absence of a core well-posedness theory for these equations is the sensitivity of their solutions to the structure of a parabolic regularization when attempting to single out an admissible solution by the vanishing viscosity approach.There is,however,theoretical and numerical evidence for the appearance of solutions that exhibit persistent oscillations,so-called oscillatory waves,which are(in general,measure-valued)solutions that emerge from Riemann data or slightly perturbed constant data chosen from the interior of the elliptic region.To capture these solutions,usually a fine computational grid is required.In this work,a version of the multiresolution method applied to a WENO scheme for systems of conservation laws is proposed as a simulation tool for the efficient computation of solutions of oscillatory wave type.The hyperbolic-elliptic 2×2 systems of conservation laws considered are a prototype system for three-phase flow in porous media and a system modeling the separation of a heavy-buoyant bidisperse suspension.In the latter case,varying one scalar parameter produces elliptic regions of different shapes and numbers of points of tangency with the borders of the phase space,giving rise to different kinds of oscillation waves.

STRONG COMPACTNESS OF APPROXIMATE SOLUTIONS TO DEGENERATE ELLIPTIC-HYPERBOLIC EQUATIONS WITH DISCONTINUOUS FLUX FUNCTION

Under a non-degeneracy condition on the nonlinearities we show that sequences of approximate entropy solutions of mixed elliptic-hyperbolic equations are strongly precompact in the general case of a Caratheodory flux vector. The proofs are based on deriving localization principles for H-measures associated to sequences of measurevalued functions. This main result implies existence of solutions to degenerate parabolic convection-diffusion equations with discontinuous flux. Moreover, it provides a framework in which one can prove convergence of various types of approximate solutions, such as those generated by the vanishing viscosity method and numerical schemes.

Convergence to the Rarefaction Wave for a Model of Radiating Gas in One-dimension

In this paper, a sequence of solutions to the one-dimensional motion of a radiating gas are con- structed. Furthermore, when the absorption coefficient a tends to oo, the above solutions converge to the rarefaction wave, which is an elementary wave pattern of gas dynamics, with a convergence rate α -1/3｜lnα｜2.

Subsonic Euler flows in a divergent nozzle

We characterize a class of physical boundary conditions that guarantee the existence and uniqueness of the subsonic Euler flow in a general finitely long nozzle.More precisely,by prescribing the incoming flow angle and the Bernoulli’s function at the inlet and the end pressure at the exit of the nozzle,we establish an existence and uniqueness theorem for subsonic Euler flows in a 2-D nozzle,which is also required to be adjacent to some special background solutions.Such a result can also be extended to the 3-D asymmetric case.