Ramón G. Plaza

Stability of radiative shock profiles for hyperbolic–elliptic coupled systems

Abstract Extending previous work with Lattanzio and Mascia on the scalar (in fluid-dynamical variables) Hamer model for a radiative gas, we show nonlinear orbital asymptotic stability of small amplitude shock profiles of general systems of coupled hyperbolic–elliptic equations of the type modeling a radiative gas, that is, systems of conservation laws coupled with an elliptic equation for the radiation flux, including in particular the standard Euler–Poisson model for a radiating gas. The method is based on the derivation of pointwise Green function bounds and description of the linearized solution operator, with the main difficulty being the construction of the resolvent kernel in the case of an eigenvalue system of equations of degenerate type. Nonlinear stability then follows in standard fashion through linear estimates derived from these pointwise bounds, combined with energy estimates of nonlinear damping type.

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 [K. Hamer, Quart. J. Mech. Appl. Math., 24 (1971), pp. 155–168], 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 the 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.

Analytical and numerical investigation of traveling waves for the Allen–Cahn model with relaxation

A modification of the parabolic Allen–Cahn equation, determined by the substitution of Fick’s diffusion law with a relaxation relation of Cattaneo–Maxwell type, is considered. The analysis concentrates on traveling fronts connecting the two stable states of the model, investigating both the aspects of existence and stability. The main contribution is the proof of the nonlinear stability of the wave, as a consequence of detailed spectral and linearized analyses. In addition, numerical studies are performed in order to determine the propagation speed, to compare it to the speed for the parabolic case, and to explore the dynamics of large perturbations of the front.

Normal Modes Analysis of Subsonic Phase Boundaries in Elastic Materials

In (1), σ = σ(U) denotes the first Piola–Kirchhoff stress and is supposed to derive from a stored-energy density function W : Rd×d + → R as σ(U) = ∂W/∂U . System (1) is hyperbolic at U if W is rank-one convex at U [Ci88], i.e., if the acoustic tensor N (U, ξ) := DW (U)(ξ, ξ) is positive definite for all ξ ∈ R. We are interested in the stability of subsonic phase boundaries, which are weak solutions to (1) of form