Search

Abstract: We consider the compressible Euler equations in three space dimensions where heat conduction is modeled by Cattaneo’s law instead of Fourier’s law. For the arising purely hyperbolic system, the asymptotic behavior of discontinuous solutions to the linearized Cauchy problem is investigated. We give a description of the behavior as time tends to infinity and, in particular, as the relaxation parameter tends to zero. The latter corresponds to the singular limit and a formal convergence to the classical (i.e. Fourier law for the heat flux–temperature relation) Euler system. We recover a phenomenon observed for hyperbolic thermoelasticity, namely the dependence of the asymptotic behavior on the mean curvature of the initial surface of discontinuity; in addition, we observe a more complex behavior in general. [Copyright &y& Elsevier]

We investigate the dispersive properties of evolution equations on waveguides with a non-flat shape. More precisely, we consider an operator with Dirichlet boundary conditions on an unbounded domain Ω, and we introduce the notion of a repulsive waveguide along the direction of the first group of variables, x. If Ω is a repulsive waveguide, we prove a sharp estimate for the Helmholtz equation Hu−λ u = f. As consequences, we prove smoothing estimates for the Schrödinger and wave equations associated to H, and Strichartz estimates for the Schrödinger equation. Additionally, we deduce that the operator H does not admit eigenvalues. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the Cauchy problem for linear and semilinear thermoelastic systems with second sound in three space dimensions with discontinuous initial data. Due to Cattaneo's law, replacing Fourier's law for heat conduction, the thermoelastic system with second sound is hyperbolic. We investigate the behavior of discontinuous solutions as the relaxation parameter tends to zero, which corresponds to a formal convergence of the system to the hyperbolic-parabolic type of classical thermoelasticity. By studying expansions with respect to the relaxation parameter of the jumps of the potential part of the system on the evolving characteristic surfaces, we obtain that the jump of the temperature goes to zero while the jumps of the heat flux and the gradient of the potential part of the elastic wave are propagated along the characteristic curves of the elastic fields when the relaxation parameter goes to zero. An interesting phenomenon is when time goes to infinity: the behavior will depend on the mean curvature of the initial surface of discontinuity. These jumps decay exponentially when time goes to infinity, more rapidly for a smaller heat conductive coefficient in linear problems and in nonlinear problems when certain growth conditions are imposed on the nonlinear functions. [ABSTRACT FROM AUTHOR]

Abstract: We consider the Timoshenko system in a bounded domain . The system has an indefinite damping mechanism, i.e. with a damping function possibly changing sign, present only in the equation for the rotation angle. We shall prove that the system is still exponentially stable under the same conditions as in the positive constant damping case, and provided and , for ϵ small enough. The decay rate will be described explicitly. In the arguments, we shall also give a new proof of exponential stability for the constant case . Moreover, we give a precise description of the decay rate and demonstrate that the system has the spectrum determined growth (SDG) property, i.e. the type of the induced semigroup coincides with the spectral bound for its generator. [Copyright &y& Elsevier]

Abstract: We consider initial–boundary value problems for the equations of isotropic elasticity for several mixed boundary conditions in infinite wave guides, as well as Maxwell equations. With the help of decompositions of the displacement field into divergence- and curl-free parts, respectively, which are compatible with the boundary conditions, we obtain sharp decay rates for the solutions. The decomposed systems correspond to the second-order Maxwell equations for the electric and the magnetic field with electric and magnetic boundary conditions, respectively. [Copyright &y& Elsevier]

We consider the Cauchy problem for a coupled system generalizing the thermoelastic plate equations. First we prove resolvent estimates for the stationary operator and conclude the analyticity of the associated semigroup in Lp-spaces, 1 < p < 1 ∞, for certain values of the parameters of the system; here the Newton polygon method is used. Then we prove decay rates of the Lq(ℝn)-norms of solutions, 2 ≥ q ≥ ∞, as time tends to infinity. [ABSTRACT FROM AUTHOR]

We investigate coupled systems of thermoelastic type in a general abstract form both modeling Fourier and Cattaneo type heat conduction. In particular we take into account a possible inertial term. A complete picture of the regions of exponential stability resp. non-exponential stability for the arising parameters (two from the type of thermoelastic system, one from the inertial term) is given. The regions of loss of exponential stability, while moving from the Fourier to the Cattaneo law, are thus clearly recognized and interestingly large. The polynomial stability in regions of non-exponential stability is also characterized. [ABSTRACT FROM AUTHOR]

For a class of bounded reference configurations that are of partially rectangular type, the linear system of magneto-elasticity in two space dimensions is considered and a polynomial decay rate of the energy as time tends to infinity is proved. [ABSTRACT FROM AUTHOR]

Abstract: Several thermomechanical models have been proposed from a heuristic point of view. A mathematical analysis should help to clarify the applicability of these models, among those recent thermal or viscoelastic models. Single-phase-lag and dual-phase-lag heat conduction models can be interpreted as formal expansions of delay equations. The delay equations are shown to be ill-posed, as are the formal expansions of higher order — in contrast to lower-order expansions leading to Fourier’s or Cattaneo’s law. The ill-posedness is proved, showing the lack of continuous dependence on the data, and thus showing that these models (delay or higher-order expansion ones) are highly explosive. In this note we shall present conditions for when this happens. [Copyright &y& Elsevier]