Showing papers in "Archive for Rational Mechanics and Analysis in 1999"

Entropy Solutions for Nonlinear Degenerate Problems

Abstract: . We consider a class of elliptic‐hyperbolic degenerate equations $$g(u)-\Delta b(u) +\divg\phi (u) =f$$ with Dirichlet homogeneous boundary conditions and a class of elliptic‐parabolic‐hyperbolic degenerate equations $$g(u)_t-\Delta b(u) +\divg\phi (u) =f$$ with homogeneous Dirichlet conditions and initial conditions. Existence of entropy solutions for both problems is proved for nondecreasing continuous functions g and b vanishing at zero and for a continuous vectorial function φ satisfying rather general conditions. Comparison and uniqueness of entropy solutions are proved for g and b continuous and nondecreasing and for φ continuous.

Divergence‐Measure Fields and Hyperbolic Conservation Laws

Abstract: We analyze a class of L 1 vector elds, called divergence-measure elds. We establish the Gauss-Green formula, the normal traces over subsets of Lipschitz boundaries, and the product rule for this class of L 1 elds. Then we apply this theory to analyzeL 1 entropy solutions of initial-boundary-value problems for hyperbolic conservation laws and to study the ways in which the solutions assume their initial and boundary data. The examples of conservation laws include multidimensional scalar equations, the system of nonlinear elasticity, and a class ofm m systems with afne characteristic hypersurfaces. The analysis inL 1 also extends toL p .

Global nonexistence theorems for a class of evolution equations with dissipation

Abstract: .We study abstract evolution equations with nonlinear damping terms and source terms, including as a particular case a nonlinear wave equation of the type \( \ba{cl} u_{tt}-\Delta u+ b|u_t|^{m-2}u_t=c|u|^{p-2}u, & (t,x)\in [0,T)\times\Omega,\\[6pt] u(t,x)=0, & (t,x)\in [0,T)\times\partial \Omega,\\[6pt] u(0,\cdot)=u_0\in H_0^1(\Omega), \quad u_t(0,\cdot)=v_0\in L^2(\Omega),\es& \ea \) where \( 0$, $p>2$, $m>1\). We prove a global nonexistence theorem for positive initial value of the energy when \( 1 1\).

Existence of Weak Solutions for the Motion of Rigid Bodies in a Viscous Fluid

Abstract: . We study the evolution of a finite number of rigid bodies within a viscous incompressible fluid in a bounded domain of \(\R^d$ $(d=2$ or $3)\) with Dirichlet boundary conditions. By introducing an appropriate weak formulation for the complete problem, we prove existence of solutions for initial velocities in \(H^1_0(\Omega)\). In the absence of collisions, solutions exist for all time in dimension 2, whereas in dimension 3 the lifespan of solutions is infinite only for small enough data.

The ∞-Eigenvalue Problem

Abstract: . The Euler‐Lagrange equation of the nonlinear Rayleigh quotient \( \left(\int_{\Omega}| abla u|^{p}\,dx\right) \bigg/ \left(\int_{\Omega}|u|^{p}\,dx\right)\) is \( -\div\left( | abla u|^{p-2} abla u \right)= \Lambda_{p}^{p} |u |^{p-2}u,\) where \(\Lambda_{p}^{p}\) is the minimum value of the quotient. The limit as \(p\to\infty\) of these equations is found to be \(\max \left\{ \Lambda_{\infty}-\frac{| abla u(x)|}{u(x)},\ \ \Delta_{\infty}u(x)\right\}=0,\) where the constant \(\Lambda_{\infty}=\lim_{p\to\infty}\Lambda_{p}\) is the reciprocal of the maximum of the distance to the boundary of the domain Ω.

A Discrete Convolution Model¶for Phase Transitions

Abstract: We study a discrete convolution model for Ising-like phase transitions. This nonlocal model is derived as an l 2-gradient flow for a Helmholtz free energy functional with general long range interactions. We construct traveling waves and stationary solutions, and study their uniqueness and stability. In particular, we find some criteria for “propagation” and “pinning”, and compare our results with those for a previously studied continuum convolution model.

L1-stability estimates for n x n conservation laws

Abstract: . Let $u_t+f(u)_x=0$ be a strictly hyperbolic $n\times n$ system of conservation laws, each characteristic field being linearly degenerate or genuinely nonlinear. In this paper we explicitly define a functional $\Phi=\Phi(u,v)$ , equivalent to the $\L^1$ distance, which is “almost decreasing” i.e., $$ \Phi\big( u(t),~v(t)\big)-\Phi\big( u(s),~v(s)\big)\leq \O(\ve)\cdot (t-s)\quad\hbox{for all}~~t>s\geq 0,$$ for every pair of e-approximate solutions u, v with small total variation, generated by a wave front tracking algorithm. The small parameter e here controls the errors in the wave speeds, the maximum size of rarefaction fronts and the total strength of all non-physical waves in u and in v. From the above estimate, it follows that front-tracking approximations converge to a unique limit solution, depending Lipschitz continuously on the initial data, in the ${\vec L}^1$ norm. This provides a new proof of the existence of the standard Riemann semigroup generated by a n×n system of conservation laws.

Fluid Films with Curvature Elasticity

Abstract: The phenomenology of surfactant fluid-film microstructures interspersed in bulk fluids poses significant challenges to continuum theory. By using simple models of elastic surfaces, chemical physicists have been partially successful in describing the qualitative features of ...

Variational formulation of softening phenomena in fracture mechanics. The one-dimensional case

Abstract: . We show that discrete models of atoms subject to nearest‐neighbour non‐linear interactions approximate continua allowing for softening and fracture. A detailed study of local minima and stationary points is carried out. Scale effects are discussed.

Decay rates for the three-dimensional linear system of thermoelasticity

Abstract: . We consider the two and three‐dimensional system of linear thermoelasticity in a bounded smooth domain with Dirichlet boundary conditions. We analyze whether the energy of solutions decays exponentially uniformly to zero as \(t\to\infty\). First of all, by a decoupling method, we reduce the problem to an observability inequality for the Lame system in linear elasticity and more precisely to whether the total energy of the solutions can be estimated in terms of the energy concentrated on its longitudinal component. We show that when the domain is convex, the decay rate is never uniform. In fact, the lack of uniform decay holds in a more general class of domains in which there exist rays of geometric optics of arbitrarily large length that are always reflected perpendicularly or almost tangentially on the boundary. We also show that, in three space dimensions, the lack of uniform decay may also be due to a critical polarization of the energy on the transversal component of the displacement. In two space dimensions we prove a sufficient (and almost necessary) condition for the uniform decay to hold in terms of the propagation of the transversal characteristic rays, under the further assumption that the boundary of the domain does not have contacts of infinite order with its tangents. We also give an example, due to D. Hulin, in which these geometric properties hold. In three space dimensions we indicate (without proof) how a careful analysis of the polarization of singularities may lead to sharp sufficient conditions for the uniform decay to hold. In two space dimensions we prove that smooth solutions decay polynomially in the energy space to a finite‐dimensional subspace of solutions except when the domain is a ball or an annulus. Finally we discuss some closely related controllability and spectral issues.

Symmetry of Ground States of Quasilinear Elliptic Equations

Abstract: . We consider the problem of radial symmetry for non‐negative continuously differentiable weak solutions of elliptic equations of the form \( {\rm div}(A(\vert Du\vert)Du) + f(u) = 0,\quad x\in {\vec R}^n, \quad n\geq 2,\eqno(1)\) under the ground state condition \( u(x)\to 0 \mbox{ as } \vert x\vert\to\infty. \eqno(2)\) Using the well‐known moving plane method of Alexandrov and Serrin, we show, under suitable conditions on A and f, that all ground states of (1) are radially symmetric about some origin O in \({\vec R}^n\). In particular, we obtain radial symmetry for compactly supported ground states and give sufficient conditions for the positivity of ground states in terms of the given operator A and the nonlinearity f.

Convergence to Travelling Fronts of Solutions of the p-System with Viscosity in the Presence of a Boundary

Abstract: . We study the asymptotic behavior as time goes to infinity of solutions to the initial‐boundary‐value problem on the half space \(R_+\) for a one‐dimensional model system for the isentropic flow of a compressible viscous gas, the so‐called p‐system with viscosity. As boundary conditions, we prescribe the constant state at infinity and require that the velocity be zero at the boundary \(x=0\). When the velocity at infinity is negative and satisfies a condition on the magnitude, we prove that if the initial data are suitably close to those for the corresponding outgoing viscous shock profile, which is suitably far from the boundary, then a unique solution exists globally in time and tends toward the properly shifted viscous shock profile as the time goes to infinity. The proof is given by an elementary energy method.

Gas Dynamics in Thermal Nonequilibrium¶and General Hyperbolic Systems¶with Relaxation

Abstract: We study gas flow in vibrational nonequilibrium. The model is a 4×4 nonlinear hyperbolic system with relaxation. Under physical assumptions, properties of thermodynamic variables relevant to stability are obtained, global existence for Cauchy problems with smooth and small data is established, and large time behavior is studied in the pointwise sense. We formulate the fundamental solution in a systematic way for a general linear system with relaxation. The fundamental solution provides insights to the behavior of the nonlinear system, and is crucial to obtain our pointwise asymptotic picture for the nonequilibrium flow. We also clarify in a general setting the relation between subcharacteristic conditions and a dissipative criterion that was originally proposed for hyperbolic-parabolic systems and has now proved to be important also for hyperbolic systems with relaxation.

An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle

Abstract: We consider the three-dimensional Navier-Stokes initial value problem in the exterior of a rotating obstacle. It is proved that a unique solution exists locally in time if the initial velocity possesses the regularity L 1/2. This regularity assumption is the same as that in the famous paper of Fujita & Kato. An essential step for the proof is the deduction of a certain smoothing property together with estimates near t≡0 of the semigroup, which is not an analytic one, generated by the operator in the space L 2, where ω stands for the angular velocity of the rotating obstacle and P denotes the projection associated with the Helmholtz decomposition.

Stable steady states in stellar dynamics.

Abstract: . We prove the existence and nonlinear stability of steady states of the Vlasov‐Poisson system in the stellar dynamics case. The steady states are obtained as minimizers of an energy‐Casimir functional from which fact their dynamical stability is deduced. The analysis applies to some of the well‐known polytropic steady states, but it also considerably extends the class of known steady states.

Symmetry of Ground States of p-Laplace Equations via the Moving Plane Method

Abstract: . In this paper we use the moving plane method to get the radial symmetry about a point \(x_0 \in {\mathbb R}^N\) of the positive ground state solutions of the equation \(-{\rm div}\; (|D u|^{p - 2} Du) = f(u)\) in \({\mathbb R}^N\), in the case \(1 < p < 2\). We assume f to be locally Lipschitz continuous in \((0, + \infty)\) and nonincreasing near zero but we do not require any hypothesis on the critical set of the solution. To apply the moving plane method we first prove a weak comparison theorem for solutions of differential inequalities in unbounded domains.

Heteroclinic Networks in Coupled Cell Systems

Abstract: . We give an intrinsic definition of a heteroclinic network as a flow-invariant set that is indecomposable but not recurrent. Our definition covers many previously discussed examples of heteroclinic behavior. In addition, it provides a natural framework for discussing cycles between invariant sets more complicated than equilibria or limit cycles. We allow for cycles that connect chaotic sets (cycling chaos) or heteroclinic cycles (cycling cycles). Both phenomena can occur robustly in systems with symmetry. We analyze the structure of a heteroclinic network as well as dynamics on and near the network. In particular, we introduce a notion of ‘depth’ for a heteroclinic network (simple cycles between equilibria have depth 1), characterize the connections and discuss issues of attraction, robustness and asymptotic behavior near a network. We consider in detail a system of nine coupled cells where one can find a variety of complicated, yet robust, dynamics in simple polynomial vector fields that possess symmetries. For this model system, we find and prove the existence of depth‐2 networks involving connections between heteroclinic cycles and equilibria, and study bifurcations of such structures.

Materials with Internal Variables and Relaxation to Conservation Laws

Abstract: .The theory of materials with internal state variables of Coleman & Gurtin [CG] provides a natural framework to investigate the structure of relaxation approximations of conservation laws from the viewpoint of continuum thermomechanics. After reviewing the requirements imposed on constitutive theories by the principle of consistency with the Clausius‐Duhem inequality, we pursue two specific theories pertaining to stress relaxation and relaxation of internal energy. They each lead to a relaxation framework for the theory of thermoelastic non‐conductors of heat, equipped with globally defined “entropy” functions for the associated relaxation process. Next, we consider a semilinear model problem of stress relaxation. We discuss uniform stability and compactness for solutions of the relaxation system in the zero‐relaxation limit, and establish convergence to the system of isothermal elastodynamics by using compensated compactness. Finally, we prove a strong dissipation estimate for the relaxation approximations proposed in Jin & Xin [JX] when the limit system is equipped with a strictly convex entropy.

Existence and Uniqueness of the Flow of Second‐Grade Fluids with Slip Boundary Conditions

Abstract: . This paper treats the solvability of the equations of motion for an incompressible fluid of grade two subject to nonlinear partial slip boundary conditions in a bounded simply‐connected domain. The existence of a unique classical solution, global in time, is proved under suitable regularity and growth restrictions on the initial data, the slip law and the body and surface forces. The method is based on a fixed-point formulation of the problem.

Decay of Entropy Solutions of Nonlinear Conservation Laws

Abstract: We are concerned with the asymptotic behavior of entropy solutions of nonlinear conservation laws. The main objective of this paper is to present an analytical approach and to explore its applications to studying the large-time behavior of periodic entropy solutions of hyperbolic conservation laws. The asymptotic decay of periodic solutions of nonlinear hyperbolic conservation laws is an important nonlinear phenomenon. It is observed that the genuine nonlinearity of equations forces the waves of each characteristic family to interact vigorously and to cancel each other. The insightful analysis of Glimm-Lax [GL], for scalar equations and 2 × 2 systems, has indicated that the resultant mutual cancellation of interacting shock and rarefaction waves of the same family induces the decay of periodic solutions. Such a result was first shown by Lax [L1] in 1957 for one-dimensional convex scalar conservation laws. Dafermos [D1], applying his uniform processes, proved the decay result for the case that the set of inflection points of the flux does not have an accumulation point. The Glimm-Lax theory [GL] indicates that, for 2× 2 strictly hyperbolic and genuinely nonlinear systems, any periodic Glimm solution decays like O(1/t). This result was proved by using the approximate characteristic method in the Glimm difference solutions, provided that the oscillation of the corresponding initial data is small. Recently, using the method of generalized characteristics, Dafermos [D3] showed that any periodic solution with local bounded variation and small oscillation for 2 × 2 systems decays asymptotically, with the same detailed structure pas found by Lax [L1] for the

Stable Configurations in Superconductivity: Uniqueness, Multiplicity, and Vortex‐Nucleation

Abstract: . We find new stable solutions of the Ginzburg‐Landau equation for high κ superconductors with exterior magnetic field h ex. First, we prove the uniqueness of the Meissner‐type solution. Then, we prove, in the case of a disc domain, the coexistence of branches of solutions with n vortices of degree one, for any n not too high and for a certain range of h ex; and describe these branches. Finally, we give an estimate on the nucleation energy barrier, to pass continuously from a vortexless configuration to a configuration with a centered vortex.

Zero-Dissipation Limit of Solutions with Shocks for Systems of Hyperbolic Conservation Laws

Abstract: We consider the convergence of solutions of conservation laws with viscosity to solutions having shocks of hyperbolic conservation laws without viscosity as the viscosity tends to zero. Our analysis reveals a rich structure of nonlinear wave interactions due to the presence of shocks and initial layers. These interactions generate four different wave patterns: initial layers, shock layers, diffusion waves and coupling waves. We study the propagation and interactions of the four wave patterns by a detailed pointwise analysis.

Derivation of the LSW‐Theory for Ostwald Ripening by Homogenization Methods

Abstract: . The coarsening of a solid phase in an undercooled liquid is described by a Stefan problem with surface tension. The solid phase is characterized by a large number of balls with small volume fraction and small capacity. We solve the quasi‐static equation as well as the parabolic problem and construct approximate solutions by means of comparison principles. The framework of Young measures is used to pass to the limit of infinitely many particles. We obtain that the capacity is the crucial quantity to get the conservation law for the particle size distribution which is studied in the classical theory for Ostwald ripening by Lifshitz, Slyozov & Wagner (LSW).

Variational Method¶for Stable Polytropic Galaxies

Abstract: It is shown that polytropic galaxies which obey the generalized Emden-Fowler law are minimizers of appropriate Energy-Casimir functionals. They are dynamically stable under spherically symmetric perturbations.

Stability of Subsonic Planar Phase Boundaries in a van der Waals Fluid

Abstract: We are concerned with the structural stability of dynamic phase changes occurring across sharp interfaces in a multidimensional van der Waals fluid. Such phase transitions can be viewed as propagating discontinuities. However, they are usually subsonic, and thus undercompressive. The lacking information lies in an additional jump condition, which may be derived from the viscosity-capillarity criterion. This condition is rather simple in the case of reversible phase transitions, since it reduces to a generalized equal area rule. In a previous work, I proved that reversible planar phase boundaries are weakly linearly stable, in the sense introduced by Majda for shock fronts. This means that they satisfy a generalized Lopatinsky condition but not a uniform one. The aim of this paper is to point out the influence of viscosity on the stability analysis, in order to deal with the more realistic case of dissipative phase transitions. The main difficulty lies in the additional jump condition, which is no longer explicit and depends on the (unknown) internal structure of the interface. We overcome it by using bifurcation arguments on the nondimensional parameter measuring the competition between viscosity and capillarity. We show by perturbation that the positivity of this parameter stabilizes the phase transitions. As a conclusion, we find that dissipative planar phase boundaries are uniformly linearly stable, in the sense of the uniform Lopatinsky condition.

A multiplicity result for solitary gravity-capillary waves in deep water via critical-point theory

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Cauchy fluxes associated with tensor fields having divergence measure

Abstract: Cauchy fluxes induced by locally summable tensor fields with divergence measure are characterized The equivalence between integral formulations involving subsets of finite perimeter and much more restricted classes of subsets is proved