Showing papers in "Archive for Rational Mechanics and Analysis in 2004"

Isotropic Hypoellipticity and Trend to Equilibrium for the Fokker-Planck Equation with a High-Degree Potential

Abstract: We consider the Fokker-Planck equation with a confining or anti-confining potential which behaves at infinity like a possibly high-degree homogeneous function. Hypoellipticity techniques provide the well-posedness of the weak Cauchy problem in both cases as well as instantaneous smoothing and exponential trend to equilibrium. Lower and upper bounds for the rate of convergence to equilibrium are obtained in terms of the lowest positive eigenvalue of the corresponding Witten Laplacian, with detailed applications.

Entropy and Global Existence for Hyperbolic Balance Laws

Abstract: This paper presents a general result on the existence of global smooth solutions to hyperbolic systems of balance laws in several space variables. We propose an entropy dissipation condition and prove the existence of global smooth solutions under initial data close to a constant equilibrium state. In addition, we show that a system of balance laws satisfies a Kawashima condition if and only if its first-order approximation, namely the hyperbolic-parabolic system derived through the Chapman-Enskog expansion, satisfies the corresponding Kawashima condition. The result is then applied to Bouchut’s discrete velocity BGK models approximating hyperbolic systems of conservation laws.

Regularity Theory for the Spatially Homogeneous Boltzmann Equation with Cut-Off

Abstract: We develop the regularity theory of the spatially homogeneous Boltzmann equation with cut-off and hard potentials (for instance, hard spheres), by (i) revisiting the Lp theory to obtain constructive bounds, (ii) establishing propagation of smoothness and singularities, (iii) obtaining estimates on the decay of the singularities of the initial datum Our proofs are based on a detailed study of the “regularity of the gain operator” An application to the long-time behavior is presented

Long-Time Asymptotics of Kinetic Models of Granular Flows

Abstract: We analyze the long-time asymptotics of certain one-dimensional kinetic models of granular flows, which have been recently introduced in [22] in connection with the quasi-elastic limit of a model Boltzmann equation with dissipative collisions and variable coefficient of restitution. These nonlinear equations, classified as nonlinear friction equations, split naturally into two classes, depending on whether or not the temperature of their similarity solutions (homogeneous cooling states) reduce to zero in finite time. For both classes, we show uniqueness of the solution by proving decay to zero in the Wasserstein metric of any two solutions with the same mass and mean velocity. Furthermore, if the temperature of the similarity solution decays to zero in finite time, we prove, by computing explicitly upper bounds for the lifetime of the solution in terms of the length of the support, that the temperature of any other solution with initially bounded support must also decay to zero in finite time.

Stability of Large-Amplitude Viscous Shock Profiles of Hyperbolic-Parabolic Systems

Abstract: We establish nonlinear L1∩H3→Lp orbital stability, 2≦p≤∞, with sharp rates of decay, of large-amplitude Lax-type shock profiles for a class of symmetric hyperbolic-parabolic systems including compressible gas dynamics and magnetohydrodynamics (MHD) under the necessary conditions of strong spectral stability, i.e., a stable point spectrum of the linearized operator about the wave, transversality of the profile, and hyperbolic stability of the associated ideal shock. This yields in particular, together with the spectral stability results of [50], the nonlinear stability of arbitrarily large-amplitude shock profiles of isentropic Navier–Stokes equations for a gamma-law gas as γ→1: the first complete large-amplitude stability result for a shock profile of a system with real (i.e., partial) viscosity. A corresponding small-amplitude result was established in [53, 54] for general systems of Kawashima class by a combination of ‘‘Kawashima-type’’ energy estimates and pointwise Green function bounds, where the small-amplitude assumption was used only to close the energy estimates. Here, under the mild additional assumption that hyperbolic characteristic speeds (relative to the shock) are not only nonzero but of a common sign, we close the estimates instead by use of a Goodman-type weighted norm [25, 26] designed to control estimates in the crucial hyperbolic modes.

Hydrodynamic Structure of the Augmented Born-Infeld Equations

Abstract: The Born-Infeld system is a nonlinear version of Maxwell’s equations. We first show that, by using the energy density and the Poynting vector as additional unknown variables, the BI system can be augmented as a 10×10 system of hyperbolic conservation laws. The resulting augmented system has some similarity with magnetohydrodynamics (MHD) equations and enjoys remarkable properties (existence of a convex entropy, Galilean invariance, full linear degeneracy). In addition, the propagation speeds and the characteristic fields can be computed in a very easy way, in contrast with the original BI equations. Then, we investigate several limit regimes of the augmented BI equations, by using a relative-entropy method going back to Dafermos, and recover the Maxwell equations for low fields, some pressureless MHD equations for high fields, and pressureless gas equations for very high fields.

The Stationary Navier-Stokes System in Nonsmooth Manifolds: The Poisson Problem in Lipschitz and C 1 Domains

Abstract: We consider the linearized version of the stationary Navier-Stokes equations on a subdomain Ω of a smooth, compact Riemannian manifold M. The emphasis is on regularity: the boundary of Ω is assumed to be only C1 and even Lipschitz, and the data are selected from appropriate Sobolev-Besov scales. Our approach relies on the method of boundary integral equations, suitably adapted to the variable-coefficient setting we are considering here. Applications to the stationary, nonlinear Navier-Stokes equations in this context are also discussed.

Local Existence for the FENE-Dumbbell Model of Polymeric Fluids

Abstract: We study the well-posedness of a multi-scale model of polymeric fluids. The microscopic model is the kinetic theory of the finitely extensible nonlinear elastic (FENE) dumbbell model. The macroscopic model is the incompressible non-Newton fluids with polymer stress computed via the Kramers expression. The boundary condition of the FENE-type Fokker-Planck equation is proved to be unnecessary by the singularity on the boundary. Other main results are the local existence, uniqueness and regularity theorems for the FENE model in certain parameter range.

Scalar Minimizers with Fractal Singular Sets

Abstract: Lack of regularity of local minimizers for convex functionals with non-standard growth conditions is considered. It is shown that for every ɛ>0 there exists a function aCα(Ω) such that the functional admits a local minimizer uW1,p(Ω) whose set of non-Lebesgue points is a closed set Σ with dimℋ(Σ)>N−p−ɛ, and where 1

Dissipative structure and entropy for hyperbolic systems of balance laws

Abstract: In this paper, we introduce an entropy condition for hyperbolic systems of balance laws. Under this condition, we use the Chapman-Enskog expansion to derive the corresponding viscous conservation laws. Further structural conditions are discussed in order to develop (local and global) existence theories for the balance laws and viscous conservation laws.

Towards a Unified Field Theory for Classical Electrodynamics

Abstract: In this paper we introduce a model which describes the relation of matter and the electromagnetic field from a unitarian standpoint in the spirit of ideas of Born and Infeld. In this model, based on a semilinear perturbation of Maxwell equations, the particles are finite-energy solitary waves due to the presence of the nonlinearity. In this respect the matter and the electromagnetic field have the same nature. Finite energy means that particles have finite mass and this makes electrodynamics consistent with the special relativity. We analyze the invariants of the motion of the semilinear Maxwell equations (SME) and their static solutions. In the magnetostatic case (i.e., when the electric field E = 0 and the magnetic field H does not depend on time) SME are reduced to the semilinear equation Open image in new window where ∇× denotes the curloperator, f′ is the gradient of a strictly convex smooth function f:R3→R and A:R3→R3 is the gauge potential related to the magnetic field H (H = ∇× A). Due to the presence of the curl operator, (1) is a strongly degenerate elliptic equation. Moreover, physical considerations impel f to be flat at zero (f′′(0)=0) and this fact leads us to study the problem in a functional setting related to the Orlicz space Lp+Lq. The existence of a nontrivial finite- energy solution of (1) is proved under suitable growth conditions on f. The proof is carried out by using a suitable variational framework related to the Hodge splitting of the vector field A.

Asymptotic States of a Smoluchowski Equation

Abstract: We study the high-concentration asymptotics of steady states of a Smoluchowski equation arising in the modeling of nematic liquid crystalline polymers.

Existence and Uniqueness of Time-Periodic Physically Reasonable Navier-Stokes Flow Past a Body

Abstract: Let Ω be a three-dimensional exterior domain of class C2,α, 0<α<1. Assume that a Navier-Stokes liquid is moving in Ω under the action of a body force F that is time-periodic of period T, and that the velocity of the liquid is zero at spatial infinity. In this paper we show that, if F satisfies suitable conditions, and its norm, in appropriate function spaces, is sufficiently small, there is at least one time-periodic strong solution. Furthermore, the velocity field v of such a solution decays to zero for large |x| as |x|−1 and its spatial gradient decays as |x|−2, both uniformly in time. In addition, the pressure p decays like |x|−2 and its gradient like |x|−3, for almost all t∈[0,T]. In the special case where F is time-independent, these solutions are also time-independent and coincide with Finn’s ‘‘physically reasonable’’ solutions [4]. Moreover, we show that our time-periodic solutions are unique in a very large class, namely, the class of time-periodic weak solutions satisfying the ‘‘energy inequality’’ and with corresponding pressure fields verifying mild summability conditions in Ω×[0,T].

Change of Sign of the Corrector’s Determinant for Homogenization in Three-Dimensional Conductivity

Abstract: In this paper we study the positivity of the determinant of the local electric field in a conducting composite. We know by [1] that the positivity holds true in two dimensions for any periodic structure. Using a different approach from [11] we prove that is also the case for a laminate microstructure in any dimension. However, and this is the main result of the paper, we provide an example of a two-phase three-dimensional periodic composite for which the determinant changes sign.

A priori Estimates and Existence for Elliptic Systems via Bootstrap in Weighted Lebesgue Spaces

Abstract: We present a new general method to obtain regularity and a priori estimates for solutions of semilinear elliptic systems in bounded domains. This method is based on a bootstrap procedure, used alternatively on each component, in the scale of weighted Lebesgue spaces Lpδ(Ω)=Lp(Ωδ(x) dx), where δ(x) is the distance to the boundary. Using this method, we significantly improve the known existence results for various classes of elliptic systems.

Convexity of Stokes Waves of Extreme Form

Abstract: Existence is established of a piecewise-convex, periodic, planar curve S below which is defined a harmonic function which simultaneously satisfies prescribed Dirichlet and Neumann boundary conditions on S. In hydrodynamics this corresponds to the existence of a periodic Stokes wave of extreme form which has a convex profile between consecutive stagnation points where there is a corner with a contained angle of 120°

Existence and conditional energetic stability of capillary-gravity solitary water waves by minimisation

Abstract: Firstly, the two-dimensional stationary water-wave problem is considered. Existence of capillary-gravity solitary waves is proved by minimising a functional related to Smale’s amended potential. We first establish the existence of periodic solutions of arbitrarily large periods, leading to a minimising sequence in L2(ℝ) that stays away from the boundary of the neighbourhood of 0 ∈ W2,2(ℝ) in which the analysis is carried out. With the help of the concentration-compactness principle, we then show that every minimising sequence has a subsequence that, after possible shifts in the propagation direction, converges in L2(ℝ) to a minimiser. Secondly, for the evolutionary problem, we prove that the set of minimal solitary waves as a whole is energetically conditionally stable. “Energetically” means that the distance to the set of all minimisers is defined in terms of the total energy, and “conditionally” means that we consider solutions to the evolutionary problem that do not explode instantaneously but could perhaps explode in finite time (e.g., via the explosion of another norm). We work in some bounded set in W2,2(ℝ) that contains the quiescent state and we are not interested in the fate of solutions that leave this set.

The Regularity of Critical Points of Polyconvex Functionals

Abstract: In this paper we are concerned with the question of regularity of critical points for functionals of the type eq1 We construct a smooth, strongly polyconvex eq2, and Lipschitzian weak solutions eq3 to the corresponding Euler-Lagrange system, which are nowhere C1. Moreover we show that F can be chosen in such a way that these irregular weak solutions are weak local minimisers.

Doubly Nonlinear Thin-Film Equations in One Space Dimension

Abstract: We consider a free-boundary problem for a class of fourth-order nonlinear parabolic equations which are degenerate both with respect to the unknown and to its third derivative The problem is relevant in the description of the surface-tension driven spreading of a non-Newtonian liquid over a solid surface in the “complete wetting” regime Relying solely on global and local energy estimates and on Bernis’ inequalities, we prove existence of solutions to this problem, and obtain sharp upper bounds for the propagation of their support A necessary condition for the occurrence of waiting-time phenomena is also derived

Hyperbolicity of the Nonlinear Models of Maxwell’s Equations

Abstract: We consider the class of nonlinear models of electromagnetism that has been described by Coleman & Dill [7]. A model is completely determined by its energy density W(B,D). Viewing the electromagnetic field (B,D) as a 3×2 matrix, we show that polyconvexity of W implies the local well-posedness of the Cauchy problem within smooth functions of class Hs with s>1+d/2.

Solutions of the Multiconfiguration Equations in Quantum Chemistry

Abstract: The multiconfiguration methods are the natural generalization of the well-known Hartree-Fock theory for atoms and molecules. By a variational method, we prove the existence of a minimum of the energy and of infinitely many solutions of the multiconfiguration equations, a finite number of them being interpreted as excited states of the molecule. Our results are valid when the total nuclear charge Z exceeds N−1 (N is the number of electrons) and cover most of the methods used by chemists. The saddle points are obtained with a min-max principle; we use a Palais-Smale condition with Morse-type information and a new and simple form of the Euler-Lagrange equations.

Solution of a Model Boltzmann Equation via Steepest Descent in the 2-Wasserstein Metric

Abstract: We study a model Boltzmann equation closely related to the BGK equation using a steepest-descent method in the Wasserstein metric, and prove global existence of energy-and momentum-conserving solutions. We also show that the solutions converge to the manifold of local Maxwellians in the large-time limit, and obtain other information on the behavior of the solutions. We show how the Wasserstein metric is natural for this problem because it is adapted to the study of both the free streaming and the ‘‘collisions’’.

Sharp-Interface Nematic–Isotropic Phase Transitions without Flow

Abstract: We derive a supplemental evolution equation for an interface between the nematic and isotropic phases of a liquid crystal when flow is neglected Our approach is based on the notion of configurational force As an application, we study the behavior of a spherical isotropic drop surrounded by a radially oriented nematic phase: our supplemental evolution equation then reduces to a simple ordinary differential equation admitting a closed-form solution In addition to describing many features of isotropic-to-nematic phase transitions, this simplified model yields insight concerning the occurrence and stability of isotropic cores for hedgehog defects in liquid crystals

Homogenization of Periodic Systems with Large Potentials

Abstract: We consider the homogenization of a system of second-order equations with a large potential in a periodic medium. Denoting by e the period, the potential is scaled as e−2. Under a generic assumption on the spectral properties of the associated cell problem, we prove that the solution can be approximately factorized as the product of a fast oscillating cell eigenfunction and of a slowly varying solution of a scalar second-order equation. This result applies to various types of equations such as parabolic, hyperbolic or eigenvalue problems, as well as fourth-order plate equation. We also prove that, for well-prepared initial data concentrating at the bottom of a Bloch band, the resulting homogenized tensor depends on the chosen Bloch band. Our method is based on a combination of classical homogenization techniques (two-scale convergence and suitable oscillating test functions) and of Bloch waves decomposition.

Symmetric Nonbarotropic Flows with Large Data and Forces

Abstract: We prove the global existence of weak solutions of the Navier-Stokes equations of compressible, nonbarotropic flow in three space dimensions with initial data and external forces which are large, discontinuous, and spherically or cylindrically symmetric. The analysis allows for the possibility that a vacuum state emerges at the origin or axis of symmetry, and the equations hold in the sense of distributions in the set where the density is positive. In addition, the mass and momentum equations hold weakly in the entire space-time domain, but with a nonstandard interpretation of the viscosity terms as distributions. Solutions are obtained as limits of solutions in annular regions between two balls or cylinders, and the analysis allows for the possibility that energy is absorbed into the origin or axis, and is lost in the limit as the inner radius goes to zero.

Symmetry-Breaking Bifurcations of Charged Drops

Abstract: It has been observed experimentally that an electrically charged spherical drop of a conducting fluid becomes nonspherical (in fact, a spheroid) when a dimensionless number X inversely proportional to the surface tension coefficient γ is larger than some critical value (i.e., when γ γ2, that is, the solution to the system of fluid equations coupled with the equation for the electrostatic potential created by the charged drop converges to the spherical solution as tρ∞ provided the initial drop is nearly spherical. We finally show that the part of the bifurcation branch at γ=γ2 which gives rise to oblate spheroids is linearly stable, whereas the part of the branch corresponding to prolate spheroids is linearly unstable.

Rotating Fluids with Self-Gravitation in Bounded Domains

Abstract: In this paper, we study the steady solutions of Euler-Poisson equations in bounded domains with prescribed angular velocity. This models a rotating Newtonian star consisting of a compressible perfect fluid with given equation of state P=eSργ. When the domain is a ball and the angular velocity is constant, we obtain both existence and non-existence theorems, depending on the adiabatic gas constant γ. In addition we obtain some interesting properties of the solutions; e.g., monotonicity of the radius of the star with both angular velocity and central density. We also prove that the radius of a rotating spherically symmetric star, with given constant angular velocity and constant entropy, is uniformly bounded independent of the central density. This is physically striking and in sharp contrast to the case of the non-rotating star. For general domains and variable angular velocities, both an existence result for the isentropic equations of state and non-existence result for the non-isentropic equation of state are also obtained.

Boundary Vortices for Thin Ferromagnetic Films

Abstract: We consider a simplified version of the micromagnetic energy for ferromagnetic samples in the shape of thin films. We study (a) stationary, stable critical points, and (b) solutions of the corresponding Landau-Lifshitz equation under a stability condition. We determine the asymptotic behaviour of solutions of these variational problems in the thin film limit. A characteristic property of the limit is the development of Ginzburg-Landau-type vortices at the boundary.

Stability of Poisson Equilibria and Hamiltonian Relative Equilibria by Energy Methods

Abstract: We develop a general stability theory for equilibrium points of Poisson dynamical systems and relative equilibria of Hamiltonian systems with symmetries, including several generalisations of the Energy-Casimir and Energy-Momentum Methods. Using a topological generalisation of Lyapunov’s result that an extremal critical point of a conserved quantity is stable, we show that a Poisson equilibrium is stable if it is an isolated point in the intersection of a level set of a conserved function with a subset of the phase space that is related to the topology of the symplectic leaf space at that point. This criterion is applied to generalise the energy-momentum method to Hamiltonian systems which are invariant under non-compact symmetry groups for which the coadjoint orbit space is not Hausdorff. We also show that a G-stable relative equilibrium satisfies the stronger condition of being A-stable, where A is a specific group-theoretically defined subset of G which contains the momentum isotropy subgroup of the relative equilibrium. The results are illustrated by an application to the stability of a rigid body in an ideal irrotational fluid.