Showing papers in "Archive for Rational Mechanics and Analysis in 1977"

The approach of solutions of nonlinear diffusion equations to travelling front solutions

Abstract: The paper is concerned with the asymptotic behavior as t → ∞ of solutions u(x, t) of the equation ut—uxx—∞;(u)=O, x∈(—∞, ∞) , in the case ∞(0)=∞(1)=0, ∞′(0)<0, ∞′(1)<0. Commonly, a travelling front solution u=U(x-ct), U(-∞)=0, U(∞)=1, exists. The following types of global stability results for fronts and various combinations of them will be given.

Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity

Abstract: Existence, uniqueness and well-posedness for a general class of quasi-linear evolution equations on a short time interval are established. These results, generalizing those of [29], are applied to second-order quasi-linear hyperbolic systems on IR" whose solutions (u(t),il(t)) lie in the Sobolev space H^(s+1) x H^5. Our results improve existing theorems by lowering the required value of s to s > (n/2) + 1, or s > n/2 in case the coefficients of the highest order terms do not involve derivatives of the unknown, and by establishing continuous dependence on the initial data for these values. As consequences we obtain well-posedness of the equations of elastodynamics if s>2.5 and of general relativity if s>l.5; s>3 was the best known previous value for systems of the type occuring in general relativity ([12], [16], [23]).

A regularity condition at the boundary for solutions of quasilinear elliptic equations

Abstract: In this paper we are concerned with the behavior at the boundary of weak solutions of the Dirichlet problem for quasilinear elliptic equations of second order in an open set O ~ IR". The main result is that a bounded weak solution is continuous at a boundary point of f2 provided that the complement of f2 in a neighborhood of this point is sufficiently " thick" when measured by an appropriate capacity. In the case of Laplace's equation this condition reduces to that considered by WIENER [W1], [W2"]. The equations considered are of the general divergence structure type

Periodic solutions of difference-differential equations

Abstract: The existence theorem of R. Nussbaum for periodic solutions of difference-differential equations is generalized to equations with a damping term. The study of such equations is motivated by recent theories of neural interactions in certain compound eyes.

Empirical foundation and axiomatic treatment of non-equilibrium temperature

Abstract: On the basis of well known empirems of classical thermodynamics, such as the First and Zeroth Laws, which lead to the definition of calorimeter systems by which heat exchanges can be measured, thermal interaction between thermally homogeneous systems is investigated by methods of set theory. In connexion with Carnap's axiom for a classical measuring quantity a thermodynamic analogue of the thermostatic temperature can be developed by generalized empirems which govern the thermal interaction between non-equilibrium and thermally homogeneous systems. An empirical identification of the contact temperature and its field formulation are given.

Superfluid mechanics for a high density of vortex lines

Abstract: There are two well known theories to describe the motion and thermodynamics of superfluids when a large number of quantized vortex lines are present and when the phenomena under study are on scales large compared with the vortex line spacing. These works have been criticised on the grounds that their governing equations for the smoothly varying, spatially averaged, fields do not satisfy the accepted invariance principles basic to modern continuum mechanics. This paper demonstrates one way in which such theories can arise from a properly invariant continuum approach and indicates the presence of hitherto unconsidered terms that bring them closer to the generally accepted microscopic picture. The resulting theory has applications both to rotating helium II in the laboratory, and to rotating neutron stars (pulsars).

Properties of solutions of an equation in the theory of infiltration

Abstract: Consider the flow of water through a homogeneous isotropic rigid porous medium. Let 0 denote the volumetric moisture content, which is defined locally as the volume of water present per unit volume of the porous medium, and let q denote the seepage velocity which is defined, also locally, as the volume of water flowing across unit area of the porous medium per unit time. Then, if the density of the water is assumed to be constant, the motion is governed by the continuity equation d0 Ot F-div q = 0 (1.1)

A Theory of Composites Modeled as Interpenetrating Solid Continua

Abstract: The differential equations and boundary conditions describing the behavior of a finitely deformable, heat-conducting composite material are derived by means of a systematic application of the laws of continuum mechanics to a well-defined macroscopic model consisting of interpenetrating solid continua. Each continuum represents one identifiable constituent of the N-constituent composite. The influence of viscous dissipation is included in the general treatment. Although the motion of the combined composite continuum may be arbitrarily large, the relative displacement of the individual constituents is required to be infinitesimal in order that the composite not rupture. The linear version of the equations in the absence of heat conduction and viscosity is exhibited in detail for the case of the two-constituent composite. The linear equations are written for both the isotropic and transversely isotropic material symmetries. Plane wave solutions in the isotropic case reveal the existence of high-frequency (optical type) branches as well as the ordinary low-frequency (acoustic type) branches, and all waves are dispersive. For the linear isotropic equations both static and dynamic potential representations are obtained, each of which is shown to be complete. The solutions for both the concentrated ordinary body force and relative body force are obtained from the static potential representation.

Asymptotic behaviour of solutions of a nonlinear diffusion equation

Abstract: where t and x denote, respectively, a time and a space coordinate, and the subscripts t and x denote partial differentiation with respect to these variables. The diffusion coefficient D(s) is defined on IR; it will be assumed that De C 2 + ' (R) (0 A > 0 for all s eX. We shall discuss two problems. I. The Cauchy problem in the strip S T = ( ~ , oo) • (0, T], where T is some fixed positive number, which eventually will tend to infinity. At t = 0 we prescribe

A thermodynamic theory of fluids and solids in electromagnetic fields

Abstract: A thermodynamic theory for the description of a polarizable and magnetizable fluid and solid is presented in which the absolute temperature is not a primitive but a derived quantity. The electromagnetic field equations and the electromechanical interaction terms are based on the so-called Chu-formulation, in which magnetization is based on a dipole model. It is shown in this theory that entropy flux equals heat flux divided by the absolute temperature. Furthermore, the Gibbs relation is a proved consequence of the application of the entropy principle adopted here. The theory is developed first for a heat conducting fluid and then for a viscous thermoelastic solid. For both, internal energy, equilibrium stress and entropy can be decomposed into two parts, one of which is due to the electromagnetic fields, while the other is the well known thermodynamic part in the absence of electromagnetic fields. The paper closes with a comparison of this theory with other thermodynamic interaction models.

Generalization of Saint-Venant's principle to micropolar continua

Abstract: The mathematical formulation and proof of Saint-Venant's principle as given by Toupin for non-polar solids is generalized to the case of micropolar elasticity. On one end of a micropolar cylinder of arbitrary length and cross-section we apply a system of self-equilibrated stresses and couple stresses. We first prove that the norms of the stress and couple stress tensors are bounded by the energy density. By means of Rayleigh's principle for the lowest natural eigenfrequency for a slice of the cylinder we then prove that the energy, stored in the cylinder beyond a certain distance from the loaded end, has an exponential decrease with this distance, thus establishing Saint-Venant's principle for the system.