Showing papers in "Archive for Rational Mechanics and Analysis in 1992"

Reduction of some classical non-holonomic systems with symmetry

Abstract: Two types of nonholonomic systems with symmetry are treated: (i) the configuration space is a total space of a G-principal bundle and the constraints are given by a connection; (ii) the configuration space is G itself and the constraints are given by left-invariant forms. The proofs are based on the method of quasicoordinates. In passing, a derivation of the Maurer-Cartan equations for Lie groups is obtained. Simple examples are given to illustrate the algorithmical character of the main results.

Viscous limits for piecewise smooth solutions to systems of conservation laws

Abstract: In this paper we study the zero dissipation problem for a general system of conservation laws with positive viscosity. It is shown that if the solution of the problem with zero viscosity is piecewise smooth with a finite number of noninteracting shocks satisfying the entropy condition, then there exist solutions to the corresponding system with viscosity that converge to the solutions of the system without viscosity away from shock discontinuities at a rate of order e as the viscosity coefficient e goes to zero. The proof uses a matched asymptotic analysis and an energy estimate related to the stability theory for viscous shock profiles.

Variational Identities and Applications to Differential Systems

Abstract: J. SERRIN 1. Introduction In the work of POnOs [P] and in recent work of Pucci & SERRIN [PS], variational identities are discussed which are useful for solving various ques- tions about elliptic differential equations and particular elliptic systems. Here we obtain a number of further applications of these identities to systems. It was noted in [PS] that these identities are closely related to Noether's Theorem for variational problems. By working out the divergence-term arising in Noether's Theorem and making some simplifications, one obtains a general identity for variational problems. If a variational integral has a symmetry group G with the following infinitesimal generator

Self-accommodation in martensite

Abstract: The shape-memory effect is a phenomenon wherein an apparently plastically deformed specimen recovers all strain when heated above a critical temperature. This is observed in some crystalline solids that undergo martensitic phase transformation. The martensitic transformation is a temperature-induced, diffusionless solid-to-solid phase transformation involving a change in crystalline symmetry. Shape-memory materials are able to transform from the high-temperature austenite to the low-temperature martensite phase without any apparent change in shape. This is known as self-accommodation. Necessary and sufficient conditions that the lattice parameters of a material must satisfy for the material to form a self-accommodating microstructure are derived. The main result states that if the austenite is cubic, the material is self-accommodating if and only if the transformation is volume preserving. On the other hand, if the symmetry of the austenite is not cubic, it is not possible to construct any microstructure that is self-accommodating unless the transformation strain or the Bain strain satisfies additional, rather strict, conditions. These results show good agreement with the available experimental data. The analysis here is significantly different from previous studies because it makes no a priori assumption on the microstructure.

Existence of planar flame fronts in convective-diffusive periodic media

Abstract: We prove the existence of planar travelling wave solutions in a reaction-diffusion-convection equation with combustion nonlinearity and self-adjoint linear part in Rn, n≧1. The linear part involves diffusion-convection terms and periodic coefficients. These travelling waves have wrinkled flame fronts propagating with constant effective speeds in periodic inhomogeneous media. We use the method of continuation, spectral theory, and the maximum principle. Uniqueness and monotonicity properties of solutions follow from a previous paper. These properties are essential to overcoming the lack of compactness and the degeneracy in the problem.

Boundary layer phenomena for differential-delay equations with state-dependent time lags, I.

Abstract: In this paper we begin a study of the differential-delay equation % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0Jc9yq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepS0he9vr0-vr% 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabew7aLjqadI% hagaqbaiaacIcacaWG0bGaaiykaiabg2da9iabgkHiTiaadIhacaGG% OaGaamiDaiaacMcacqGHRaWkcaWGMbGaaiikaiaadIhacaGGOaGaam% iDaiabgkHiTiaadkhacaGGPaGaaiykaiaacYcacaqGGaGaaeiiaiaa% dkhacqGH9aqpcaWGYbGaaiikaiaadIhacaGGOaGaamiDaiaacMcaca% GGPaaaaa!5192! $$\varepsilon x'(t) = - x(t) + f(x(t - r)), r = r(x(t))$$ . We prove the existence of periodic solutions for 0<ɛ<ɛ 0, where ɛ 0 is an optimal positive number. We investigate regularity and monotonicity properties of solutions x(t) which are defined for all t and of associated functions like η(t)=t−r(x(t)). We begin the development of a Poincare-Bendixson theory and phase-plane analysis for such equations. In a companion paper these results will be used to investigate the limiting profile and corresponding boundary layer phenomena for periodic solutions as ɛ approaches zero.

On the thermodynamics of periodic phases

Abstract: A computer system has been developed to handle archiving and analysis of data acquired during operations of the Continuous Wave Deuterium Demonstrator (CWDD) Data files generated by the CWDD Instrumentation and Control system are transferred across a local area network to the CWDD Archive system where they are enlisted into the archive and stored on removeable media optical disk drives A relational database management system maintains an on-line database catalog of all archived files This database contains information about file contents and formats, and holds signal parameter configuration tables needed to extract and interpret data from the files Software has been developed to assist the selection and retrieval of data on demand based upon references in the catalog Data retrieved from the archive is transferred to commercial data visualization applications for viewing, plotting and analysis

The regularized Chapman-Enskog expansion for scalar conservation laws

Abstract: Rosenau has recently proposed a regularized version of the Chapman-Enskog expansion of hydrodynamics. This regularized expansion resembles the usual Navier-Stokes viscosity terms at law wave-numbers, but unlike the latter, it has the advantage of being a bounded macroscopic approximation to the linearized collision operator. The behavior of Rosenau regularization of the Chapman-Enskog expansion (RCE) is studied in the context of scalar conservation laws. It is shown that thie RCE model retains the essential properties of the usual viscosity approximation, e.g., existence of traveling waves, monotonicity, upper-Lipschitz continuity..., and at the same time, it sharpens the standard viscous shock layers. It is proved that the regularized RCE approximation converges to the underlying inviscid entropy solution as its mean-free-path epsilon approaches 0, and the convergence rate is estimated.

On steady distributions of self-attracting clusters under friction and fluctuations

Abstract: We consider the Vlasov-Fokker-Planek equation with a Newtonian, attracting potential and study its stationary solutions, given by the generalized Lane-Emden equation. In a two-dimensional domain we obtain the existence of a critical mass beyond which the system may admit a gravitational collapse. For a one-dimensional model we prove some results on existence, uniqueness, stability and symmetry-breaking of stationary solutions.

Initial-Boundary value problems for the Boltzmann equation: Global existence of weak solutions

Abstract: For Ω an open set of ℝ3 bounded or not, we consider initial-boundary value problems for the Boltzmann equation. For general gas-surface interaction laws and for hard potentials, we prove a global existence result for weak solutions. The proof uses the regularization of the collision operator and the renormalization method for the regularized problem. By using weak compactness in L1 and averaged stability ofQ(f,f), we prove the existence of weak solutions of our problem.

Energy minimization and the formation of microstructure in dynamic anti-plane shear

Abstract: We investigate the behavior of a continuum model designed to provide insight into the dynamical development of microstructures observed during displacive phase transformations in certain materials. The model is presented within the framework of nonlinear viscoelasticity and is also of interest as an example of a strongly dissipative infinite-dimensional dynamical system whose forward orbits need not lie on a finite-dimensional attracting set, and which can display a subtle dependence on initial conditions quite different from that of classical finite-dimensional “chaos”. We study the problem of dynamical (two-dimensional) anti-plane shear with linear viscoelastic damping. Within the framework of nonlinear hyperelasticity, we consider both isotropic and anisotropic constitutive laws which can allow different phases and we characterize their ability to deliver minimizers and minimizing sequences of the stored elastic energy (Theorem 2.3). Using a transformation due to Rybka, we recast the problem as a semilinear degenerate parabolic system, thereby allowing the application of semigroup theory to establish existence, uniqueness and regularity of solutions in L p spaces (Theorem 3.1). We also discuss the issues of energy minimization and propagation of strain discontinuities. We comment on the difficulties encountered in trying to exploit the geometrical properties of specific constitutive laws. In particular, we are unable to obtain analogues of the absence of minimizers and of the non-propagation of strain discontinuities found by Ball, Holmes, James, Pego & Swart [1991] for a one-dimensional model problem. Several numerical experiments are presented, which prompt the following conclusions. It appears that the absence of an absolute minimizer may prevent energy minimization, thereby providing a dynamical mechanism to limit the fineness of observed microstructure, as has been proved in the one-dimensional case. Similarly, viscoelastic damping appears to prevent the propagation of strain discontinuities. During the extremely slow development of fine structure, solutions are observed to display local refinement in an effort to overcome incompatibility with boundary and initial conditions, with the distribution and shape of the resulting finer scales displaying a subtle dependence on initial conditions.

An example of a quasiconvex function that is not polyconvex in two dimensions

Abstract: We study the different notions of convexity for the function fγ(ξ) = |ξ|2 (|ξ|2 − 2γ det ξ) where ξ e ℝ2×2, introduced by Dacorogna & Marcellini. We show that fγ is convex, polyconvex, quasiconvex, rank-one convex, if and only if ¦γ¦≦ 2/3 √2, 1, 1+ɛ (for some ɛ>0), 2/√3, respectively.

Hyperbolic phenomena in a strongly degenerate parabolic equation

Abstract: We consider the equation u t =(ϕ(u) ψ (u x )) x , where ϕ>0 and where ψ is a strictly increasing function with lim s→∞ ψ=ψ ∞<∞. We solve the associated Cauchy problem for an increasing initial function, and discuss to what extent the solution behaves qualitatively like solutions of the first-order conservation law u t =ψ ∞ (ϕ(u)) x . Equations of this type arise, for example, in the theory of phase transitions where the corresponding free-energy functional has a linear growth rate with respect to the gradient.

On the material symmetry group of elastic crystals and the Born Rule

Abstract: The aim of this paper is to investigate the extent to which non-linear elasticity theory can be used for describing the behavior of crystalline solids. The results we obtain show that some strict and definite boundaries must be set to the possibility of actually doing so. Specifically, we address here a twofold problem: On one hand, we pose the question of the validity of the socalled “Born Rule”, a fundamental hypothesis due to Cauchy and, in a weaker form, to Born, by means of which continuum theories of crystal mechanics are formulated. On the other hand, we explore the possibility of developing an effective elastic model independently of the Rule, in those cases in which the Rule itself does not work. Our results are based on a close study of the implications of the phenomenon of mechanical twinning with regard to the symmetry properties of the energy function of an elastic crystal. These are summarized by the choice of a “material symmetry group” G: The main experimental features of twinning lead one to consider a class of “twinning subgroups” of G that are particular “reflection groups”, in fact, particular linear representations, possibly unfaithful, of abstract “Coxeter groups”. In the “generic” case, the properties of such groups prevent the Born Rule from holding. Only when some very special non-generic conditions are met by the twinning modes of a crystalline substance is the Rule valid; the development of an elastic model is then possible by following a well-known procedure. The analysis of relevant experimental data confirms that, while basically all crystals exhibit twins, most of them do exhibit generic twinning modes for which the hypothesis of Born is violated. We also show that in such generic cases any tentative thermoelastic approach developed independently of the Rule does not give physically sound results and thus cannot be usefully adopted, because some quite undesirable conclusions regarding the symmetry of the energy can be drawn that definitely make elasticity inadequate for our purposes. Experimental data point out nonetheless two quite remarkable classes of “nongeneric” materials for which the Born Rule is never violated, and to which an elastic model safely applies.

Stability of travelling fronts in a model for flame propagation Part II: Nonlinear stability

Abstract: This paper is concerned with the nonlinear stability of the travelling-wave solutions of the multidimensional thermodiffusive model for flame propagation, with unit Lewis number. The model consists in a semilinear parabolic equation in an infinite cylinder, with Neumann boundary conditions. We prove that any solution which is initially close to a travelling wave will converge to a translate of that wave.

Bifurcation analysis of minimizing harmonic maps describing the equilibrium of nematic phases between cylinders

Abstract: Consider a nematic liquid crystal confined between two coaxial circular cylinders centered on the z-axis. Suppose that each cylinder imposes a strong anchoring boundary condition requiring the director field to be normal to it. In experiments on a nematic material # in such a configuration, WILLIAMS, CLADIS, & KL~MAN [1973, 21 observed that, for the range of cylinder diameters studied, at equilibrium the director lies along radial lines in the (x,y)-plane with no component in the z-direction. We shall here show that such is the case in the one-constant theory, i.e., for minimizing harmonic maps into S 2, provided the ratio p of the radius of the inner cylinder to that of the outer cylinder is equal to or greater than e-n; however, if p is less than e -~, then the free-energy has a minimizer that preserves radial symmetry in the sense that the projection of u onto the (x ,y) -p lane lies along radial lines, but this minimizer has a component along the z-axis at every point off the bounding surfaces. It is a consequence of an elementary remark, which is here called \"Theorem 0\" and whose proof is given in Appendix I, that the present minimization problems for functions on a three-dimensional region can be reduced to problems for functions on two-dimensional regions. Let f2 be a smooth bounded domain in [E 2, put D = f2 x [0, 1], and let (0 be a given smooth map from 0f2 to S 2. Define the sets 3 a n d ~ by

Stability of travelling fronts in a model for flame propagation part I: Linear analysis

Abstract: We investigate the stability of travelling wave solutions of the multidimensional thermodiffusive model for flame propagation with unit Lewis number. This model consists in a system of two nonlinear parabolic equations posed in an infinite cylinder, with Neumann boundary conditions. In this paper, we prove that every travelling wave solution of this model is linearly stable. Our tools are exponential decay estimates for solutions of elliptic equations in a cylinder, and the Maximum Principle for parabolic equations.

Collisionless periodic solutions to some three-body problems

Abstract: We provide sufficient conditions for the existence of periodic solutions to some three-body problems. Periodic solutions are found as minima of the associated action integral and are shown to be free of double and triple collisions.