Hyperbolic Relaxation Research Articles

Relative Entropy in Hyperbolic Relaxation

We provide a framework so that hyperbolic relaxation systems are endowed with a relative entropy identity. This allows a direct proof of convergence in the regime that the limiting solution is smooth.

Global well-posedness and relaxation limits of a model for radiating gas

We study the initial value problem for a hyperbolic–elliptic coupled system with arbitrary large discontinuous initial data. We prove existence and uniqueness for that model by means of L 1-contraction and comparison properties. Moreover, after suitable scalings, we study both the hyperbolic–parabolic and the hyperbolic–hyperbolic relaxation limits for that system.

Basic structures of hyperbolic relaxation systems

This work is concerned with basic structural properties of first-order hyperbolic systems with source terms divided by a small parameter ε. We identify a relaxation criterion necessary for the solution sequences indexed with ε to have reasonable limits as ε goes to zero. This relaxation criterion is shown to imply hyperbolicity of the reduced systems governing the limits. Moreover, we introduce a so-called GC-stability theory and strengthen the hyperbolicity result. The latter shows that there are no linearly stable hyperbolic relaxation approximations for non-hyperbolic conservation laws.

Basic structures of hyperbolic relaxation systems

This work is concerned with basic structural properties of first-order hyperbolic systems with source terms divided by a small parameter ε. We identify a relaxation criterion necessary for the solution sequences indexed with ε to have reasonable limits as ε goes to zero. This relaxation criterion is shown to imply hyperbolicity of the reduced systems governing the limits. Moreover, we introduce a so-called GC-stability theory and strengthen the hyperbolicity result. The latter shows that there are no linearly stable hyperbolic relaxation approximations for non-hyperbolic conservation laws.

Pointwise Green's function bounds and stability of relaxation shocks

We establish sharp pointwise Green's function bounds and consequent linearized and nonlinear stability for smooth traveling front solutions, or relaxation shocks, of general hyperbolic relaxation systems of dissipative type, under the necessary assumptions ([G,Z.1,Z.4]) of spectral stability, i.e., stable point spectrum of the linearized operator about the wave, and hyperbolic stability of the corresponding ideal shock of the associated equilibrium system. This yields, in particular, nonlinear stability of weak relaxation shocks of the discrete kinetic Jin--Xin and Broadwell models. The techniques of this paper should have further application in the closely related case of traveling waves of systems with partial viscosity, for example in compressible gas dynamics or MHD.

Multiplication invariant subspaces of the Bergman space

We establish sharp pointwise Green's function bounds and consequent linearized stability for smooth traveling front solutions, or relaxation shocks, of general hyperbolic relaxation systems of dissipative type, under the necessary assumptions ([32, 108, 110]) of spectral stability, i.e., stable point spectrum of the linearized operator about the wave, hyperbolic stability of the corresponding ideal shock of the associated equilibrium system, and transversality of the connecting profile, with no additional assumptions on the structure or strength of the shock. Restricting to Lax type shocks, we establish the further result of nonlinear stability with respect to small L 1 n H 2 perturbations, with sharp rates of decay in L p , 2 ≤ p ≤ ∞, for weak shocks of general simultaneously symmetrizable systems; for discrete kinetic models, and initial perturbation small in W 3,1 n W 3, ∞ , we obtain sharp rates of decay in L p , 1 ≤ p < ∞, for (Lax type) shocks of arbitrary strength. This yields, in particular, nonlinear stability of weak relaxation shocks of the discrete kinetic Jin-Xin and Broad-well models, for which spectral stability has been established in [61, 43], and in [52], respectively. Our analysis follows the basic pointwise semigroup approach introduced by Zumbrun and Howard [107] for the study of traveling waves of parabolic systems; however, significant extensions are required to deal with the nonsectorial generator and more singular short-time behavior of the associated (hyperbolic) linearized equations. Our main technical innovation is a systematic method for refining large-frequency (short-time) estimates on the resolvent kernel, suitable in the absence of parabolic smoothing.

Delayed luminescence of biological systems in terms of coherent states

Delayed luminescence is the long-term “ultraweak” afterglow of biological systems after exposure to light illumination, it displays hyperbolic-like relaxation with sometimes remarkable hyperbolic oscillations around the decay function. We demonstrate that the hyperbolic relaxation of biological systems is a characteristic active response of an ergodic coherent state, while the hyperbolic-like oscillations are consequences of the coupling of coherent states with a necessary difference in the characteristic damping frequencies.

CONVERGENCE OF A RELAXATION APPROXIMATION TO A BOUNDARY VALUE PROBLEM FOR CONSERVATION LAWS

In this paper we investigate a hyperbolic relaxation approximation to the scalar conservation law for (x,t) ∈ R + × (0,T) (T > 0) and for a smooth flux function f, with the initial condition for x ...

TIME-ASYMPTOTIC STABILITY OF BOUNDARY-LAYERS FOR A HYPERBOLIC RELAXATION SYSTEM

This work is concerned with time-asymptotic stability of boundary-layers for a typical hyperbolic relaxation system. Under a nonclassical requirement characterizing a class of boundary conditions for the typical system, we prove the global (in time) existence and asymptotic decay of solutions with initial data close to the steady solutions or relaxation boundary-layers.

The Lp Stability of Relaxation Rarefaction Profiles

We consider the large time behavior of solutions for a hyperbolic relaxation system. For a certain class of initial data the solution is shown to converge to relaxation rarefaction profiles at a determined asymptotic rate. The result is established without the smallness conditions of the wave strength and the initial disturbances.

Existence of Relaxation Shock Profiles for Hyperbolic Conservation Laws

In this paper we prove that, like viscosity, relaxation can also smooth weak shocks for strict hyperbolic conservation laws as equilibium systems of hyperbolic relaxation systems under some natural structural conditions. These conditions were derived previously by Yong from stability considerations and hence are entirely analogous to the Majda--Pego criterion of the viscous case. The proof involves a new parametrization of Hugoniot curves and a delicate analysis of the algebraic structure of relaxation systems as well as construction of an appropriate center manifold.

Efficient Asymptotic-Preserving (AP) Schemes For Some Multiscale Kinetic Equations

Many kinetic models of the Boltzmann equation have a diffusive scaling that leads to the Navier--Stokes type parabolic equations as the small scaling parameter approaches zero. In practical applications, it is desirable to develop a class of numerical schemes that can work uniformly with respect to this relaxation parameter, from the rarefied kinetic regimes to the hydrodynamic diffusive regimes. An earlier approach in [S. Jin, L. Pareschi, and G. Toscani, SIAM J. Numer. Anal., 35 (1998), pp. 2405--2439] reformulates such systems into the common hyperbolic relaxation system by Jin and Xin for hyperbolic conservation laws used to construct the relaxation schemes and then uses a multistep time-splitting method to solve the relaxation system. Here we observe that the combination of the two time-split steps may yield hyperbolic-parabolic systems that are more advantageous, in both stability and efficiency, for numerical computations. We show that such an approach yields a class of asymptotic-preserving (AP) schemes which are suitable for the computation of multiscale kinetic problems. We use the Goldstein--Taylor and Carleman models to illustrate this approach.

Rapid solidification and a finite velocity for the propagation of heat

The Fourier thermal conduction model is subject to a pathological anomaly in that it predicts an infinite velocity of propagation for heat. A finite propagation velocity can be introduced by adding a hyperbolic relaxation term to the conventional model. We show that in high-speed pulsed laser melting experiments, ignoring the hyperbolic nature of thermal conduction may lead to the surface temperature being seriously underestimated. We also develop a model of dendritic growth within a hyperbolic conduction framework. It is found that stable dendritic growth is only possible if the growth velocity is less than half the thermal wave velocity, C.

Convergence of phase field to phase relaxation models with memory

This paper is concerned with phase field models accounting for memory effects and based on the linearized Gurtin-Pipkin constitutive assumption for the heat flux. After recalling and generalizing existence, uniqueness, and regularity results recently obtained by the authors, here the asymptotic behaviour of the solutions to related initial-boundary value problems is investigated as the coefficient of the interfacial energy tends to zero. It is found that the limits of suitable subsequences solve the induced hyperbolic phase relaxation problem, for which the question of uniqueness is still open.

Hyperbolic relaxation as a sufficient condition of a fully coherent ergodic field

In three cases, one originating from a classical model, the second from the time-evolution operator, and the third from photocount statistics, it is shown that an initially excited coherent field which remains coherent in time development relaxes according to a hyperbolic rather than to an exponential law. This has particular relevance for the analysis of biological systems.

Approximation of multidimensional Stefan-like problems via hyperbolic relaxation

An effective time delay in the governing equations, as proposed in [8], is further examined here. The resulting hyperbolic Stefan problem is mainly viewed as a perturbation of the traditional one. Asymptotic energy error estimates are derived for the three physical unknowns (heat flux, temperature and enthalpy) in accordance with various regularity assumptions on the initial data.

Physical aspects of biophotons.

By comparing the theoretically expected results of photon emission from a chaotic (thermal) field and those of an ordered (fully coherent) field with the actual experimental data, one finds ample indications for the hypothesis that 'biophotons' originate from a coherent field occurring within living tissues. A direct proof may be seen in the hyperbolic relaxation dynamics of spectral delayed luminescence under ergodic conditions. A possible mechanism has to be founded on Einstein's balance equation and, under stationary conditions, on energy conservation including a photochemical potential. It is shown that the considered equations deliver, besides the thermal equilibrium, a conditionally stable region far away from equilibrium, which can help to describe both 'biophoton emission' and biological regulation.

Nonlinearity of the dynamic properties of rubbers at large inhomogeneous compressive deformations

The isothermal inhomogeneous harmonic compression has been calculated for a material with nonlinear elastic characteristics and a hyperbolic relaxation time distribution. The experimental data for rubbers satisfactorily confirm the theory. The proposed characteristics of the elasto-hysteresis properties of rubbers do not depend on the test piece geometry and are functions of the degree of deformation.