Constantine M. Dafermos

Asymptotic stability in viscoelasticity

where p, c are positive constants and g(r is independent of x; u(x, t) has been assigned on [0, 1 ] x ( oo, 0]. In a recent paper [1] the author investigated the problem of asymptotic stability for a class of abstract integrodifferential equations in function space, and the results were then applied to viscoelasticity. It was established, for instance, that the solutions of (1.1) are asymptotically stable provided that g(T) is a nonnegative, monotonic non-increasing and convex function such that

Asymptotic behavior of nonlinear contraction semigroups

The asymptotic behavior of weak solutions of the equation {u(t) + Au(t) ϵ ƒ(t) (A maximal monotone in Hilbert space) is determined via a characterization of ω-limit sets of the contraction semigroup generated by −A.

The Entropy Rate Admissibility Criterion for Solutions of Hyperbolic Conservation Laws

The entropy rate admissibility criterion for solutions of hyperbolic conservation laws is numerically analyzed. The following admissibility criterion for solutions of hyperbolic conservation laws is proposed: a weak solution is admissible if the total entropy decays with the highest possible rate. The equivalence of this criterion and viscosity criterion is established for the single equation and the system of equations of one dimensional nonlinear elasticity.

Asymptotic Behavior of Solutions of Evolution Equations

Publisher Summary This chapter presents the methods of investigation of the asymptotic behavior of solutions of evolution equations, endowed with a dissipative mechanism, based on the study of the structure of the ω-limit set of trajectories of the evolution operator generated by the equation. The dissipative mechanism usually manifests itself by the presence of a Liapunov functional, which is constant on ω -limit sets; the central idea of the approach presented in the chapter is to use this information in conjunction with properties of ω -limit sets such as invariance and minimality. The chapter discusses two examples of wave equations with weak damping for which the scheme set up by Hale applies. In particular, this requires that the Liapunov functional be continuous on phase space. The chapter explores the case of a hyperbolic conservation law that generates a semigroup on space. It also presents a survey of various applications and extensions of these ideas that may serve as a guide to those interested in learning more about the method.

The mixed initial-boundary value problem for the equations of nonlinear one-dimensional viscoelasticity

It has been proved (Lax [I], MacCamy-Mizel [2]) that the equations of one-dimensional nonlinear elasticity do not admit, in general, smooth solutions in the large. It is expected though, that if the stress depends on the history of motion in an appropriate fashion, then smooth solutions exist. The simplest model of a solid with history dependence is provided by one-dimensional viscoelasticity, where the stress a is a function of the deformation gradient u, and its time derivative zi, . Greenberg, MacCamy and Mizel [3] have considered the semilinear case, cr(uZ , zi.) = ~J(zL~) + ti, where 93 is a strictly increasing function. They prove the existence of a unique solution which is asymptotically stable. In this paper we consider the traction boundary value problem in the general case where O(U, , ti,) may be nonlinear in both u, , zi, . The form of the dependence of CT(U~ , 2%) on ti, is restricted by the requirement that the viscosity be bounded away from zero. On the contrary, the dependence on U, is essentially unrestricted apart from certain requirements of boundedness. It turns out that the viscoelastic part dominates the elastic part and secures the existence of a unique solution in the large. This solution is smooth enough so that all derivatives entering the equation of motion are Holder continuous. The tools of the proof are certain “energy” estimates combined with known a priori bounds from the theory of parabolic equations, and the Leray-Schauder fixed point theorem. In the final part of the paper, we investigate the asymptotic stability of the solution. The mechanism which provokes the decay of the solution is induced by the viscosity. On the other hand, the number and the nature of all possible static configurations depend entirely on the elastic part of

Energy methods for quasilinear hyperbolic initial-boundary value problems. Applications to elastodynamics

The aim of this paper is to show how energy methods can be used to establish local (in time) existence of smooth solutions to certain initial-boundary value problems for systems of quasilinear hyperbolic partial differential equations. Our particular objective is to treat the equations of nonlinear elastodynamics.

Energy methods for nonlinear hyperbolic volterra integrodifferential equations

Abstract : We use energy methods to study global existence, boundedness, and asymptotic behavior as t approaches infinity, of solutions of the two Cauchy problems (and related initial-boundary value problems).

An invariance principle for compact processes

The classical Liapunov theory of asymptotic stability relies on the construction of a Liapunov functional with a negative definite time derivative. Nevertheless, for a wide class of evolutionary equations of mathematical and physical interest, the inherent dissipative mechanism yields “natural” Liapunov functionals with negative semidefinite time derivatives. It was pointed out by LaSalle [l] that in the case of autonomous systems of ordinary differential equations the negative definiteness condition can be weakened if one takes into account the invariance of limit sets of so1utions.l An asymptotic stability theory which utilizes the invariance principle is now available for general topological dynamical systems (Hale [3]). In view of the success of this approach, a search was conducted for uncovering cases of nonautonomous systems of ordinary differential equations endowed with an invariance principle. Generalized forms of invariance principles were established for the classes of asymptotically autonomous (Markus [4], Opial [5]), periodic (LaSalle [6]) and asymptotically almost periodic systems (Miller [7]). Sell [8] introduces the concept of “limiting equations” and deduces the above results in a systematic way. Invariance principles have also been considered for other types of evolutionary equations which do not generate dynamical systems. In this category we mention Miller’s work [9] on almost periodic functional differential equations and a forthcoming article [IO] by Slemrod on periodic dynamical systems in Banach spaces. The conceptual similarity between results concerning a wide diversity of evolutionary equations motivates the development of an abstract unifying theory. In this work we study invariance in the framework of the theory of “processes.” A process is a direct generalization of the concept of a topological dynamical system. The transition from dynamical systems to

The Riemann problem for certain classes of hyperbolic systems of conservation laws

is called a Riemann problem. The classical method of solution of the Riemann problem is based on the construction of the shock and wave curves of (1.1). However, these curves can be constructed only if the shock admissibility conditions are known a priori. For this reason, the method has been applied so far only for genuinely nonlinear systems [I, 2J and the system generated by the second order wave equation [3,4]. (For results published after the completion of this work, see [9].) Our investigation here is based on a different approach, which was intro-

A system of hyperbolic conservation laws with frictional damping

The aim of this paper is to construct admissible weak solutions, in the space BV of functions of bounded variation, for the Cauchy problem