Showing papers in "Communications on Pure and Applied Mathematics in 2007"

Regularity of the obstacle problem for a fractional power of the laplace operator

Abstract: Given a function ϕ and s ∈ (0, 1), we will study the solutions of the following obstacle problem: • u ≥ ϕ in R n , • (−� ) s u ≥ 0i nR n , • (−� ) s u(x) = 0 for those x such that u(x )>ϕ (x), • lim|x|→+∞ u(x) = 0. We show that when ϕ is C 1,s or smoother, the solution u is in the space C 1,α for every α< s. In the case where the contact set {u = ϕ} is convex, we prove the optimal regularity result u ∈ C 1,s . When ϕ is only C 1,β for a β< s, we prove that our solution u is C 1,α for every α< β. c � 2006 Wiley Periodicals, Inc.

Asymptotic speeds of spread and traveling waves for monotone semiflows with applications

Abstract: The theory of asymptotic speeds of spread and monotone traveling waves is established for a class of monotone discrete and continuous-time semiflows and is applied to a functional differential equation with diffusion, a time-delayed lattice population model and a reaction-diffusion equation in an infinite cylinder. c � 2006 Wiley Periodicals, Inc.

Linear matrix inequality representation of sets

Abstract: This article concerns the question, Which subsets of ℝm can be represented with linear matrix inequalities (LMIs)? This gives some perspective on the scope and limitations of one of the most powerful techniques commonly used in control theory. Also, before having much hope of representing engineering problems as LMIs by automatic methods, one needs a good idea of which problems can and cannot be represented by LMIs. Little is currently known about such problems. In this article we give a necessary condition that we call “rigid convexity,” which must hold for a set ⊆ ℝm in order for to have an LMI representation. Rigid convexity is proved to be necessary and sufficient when m = 2. This settles a question formally stated by Pablo Parrilo and Berndt Sturmfels in [15]. As shown by Lewis, Parillo, and Ramana [11], our main result also establishes (in the case of three variables) a 1958 conjecture by Peter Lax on hyperbolic polynomials. © 2006 Wiley Periodicals, Inc.

Infinite time aggregation for the critical Patlak-Keller-Segel model in ℝ2

Abstract: We analyze the two-dimensional parabolic-elliptic Patlak-Keller-Segel model in the whole Euclidean space R2. Under the hypotheses of integrable initial data with finite second moment and entropy, we first show local in time existence for any mass of "free-energy solutions", namely weak solutions with some free energy estimates. We also prove that the solution exists as long as the entropy is controlled from above. The main result of the paper is to show the global existence of free-energy solutions with initial data as before for the critical mass 8 Π/Χ. Actually, we prove that solutions blow-up as a delta dirac at the center of mass when t→∞ keeping constant their second moment at any time. Furthermore, all moments larger than 2 blow-up as t→∞ if initially bounded.

Mean Field Dynamics of Boson Stars

Abstract: We consider a quantum mechanical system of N bosons with relativistic dispersion interacting through a mean field Coulomb potential (attractive or repulsive). We choose the initial wave function to describe a condensate where the N bosons are all in the same one-particle state. Starting from the N-body Schrodinger equation, we prove that, in the limit N ∞, the time evolution of the one-particle density is governed by the relativistic nonlinear Hartree equation. This equation is used to describe the dynamics of boson stars (Chandrasekhar theory). The corresponding static problem was rigorously solved in [10]. © 2005 Wiley Periodicals, Inc.

On the vanishing viscosity limit for the 3D navier-stokes equations with a slip boundary condition

Abstract: where and below ∇· and ∇× denote the div and curl operators respectively, n is the outward normal, and τ is the unit tangential vector of ∂Ω. The investigation of vanishing viscosity limit of solutions of the Navier-Stokes equations both in the two and three spacial dimensional cases is a classical issue. There are two related questions arising from here: one is how to describe the inviscid limiting behavior of the Navier-Stokes equation; and the other is that does the Euler equation can be approximated by the Navier-Stokes equations. In the case that the solution to the ∗This research is supported in part by Zheng Ge Ru Foundation, and Hong Kong RGC Earmarked Research Grants CUHK-4028/04P, CUHK-4040/02P and CUHK-4279/00P.

Second-Order Backward Stochastic Differential Equations and Fully Nonlinear Parabolic PDEs

Abstract: For a d-dimensional diffusion of the form dXt = �( Xt)dt + σ(Xt)dWt and continuous functions f and g, we study the existence and uniqueness of adapted processes Y, Z, Ŵ, and A solving the second-order backward stochastic differential equation (2BSDE) dYt = f (t, Xt,Yt, Zt, Ŵt)dt + Z ′ ◦ dXt, t ∈ [0,T ), dZt = At dt + Ŵt dXt,

Metric‐based upscaling

Abstract: We consider divergence form elliptic operators in dimension n ge; 2 with L∞ coefficients. Although solutions of these operators are only Holder-continuous, we show that they are differentiable (C1, α) with respect to harmonic coordinates. It follows that numerical homogenization can be extended to situations where the medium has no ergodicity at small scales and is characterized by a continuum of scales. This new numerical homogenization method is based on the transfer of a new metric in addition to traditional averaged (homogenized) quantities from subgrid scales into computational scales. Error bounds can be given and this method can also be used as a compression tool for differential operators. © 2006 Wiley Periodicals, Inc.

Concentration on curves for nonlinear Schrödinger Equations

Abstract: We consider the problem where p > 1, e > 0 is a small parameter, and V is a uniformly positive, smooth potential. Let Γ be a closed curve, nondegenerate geodesic relative to the weighted arc length ∫ΓVσ, where σ = (p + 1)/(p − 1) − 1/2. We prove the existence of a solution uϵ concentrating along the whole of Γ, exponentially small in e at any positive distance from it, provided that e is small and away from certain critical numbers. In particular, this establishes the validity of a conjecture raised in 3 in the two-dimensional case. © 2006 Wiley Periodicals, Inc.

Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy

Abstract: We study the asymptotic time behavior of global smooth solutions to general entropy dissipative hyperbolic systems of balance law in m space dimensions, under the Shizuta-Kawashima condition. We show that these solutions approach constant equilibrium state in the Lp-norm at a rate O(t^(-m/2(1-1/p))), as t tends to $\infty$, for p in [min ( m,2),+ \infty]. Moreover, we can show that we can approximate, with a faster order of convergence, theconservative part of the solution in terms of the linearized hyperbolic operator for m >= 2, and by a parabolic equation in the spirit of Chapman-Enskog expansion. The main tool is given by a detailed analysis of the Green function for the linearized problem.

Blowup for nonlinear wave equations describing boson stars

Abstract: We consider the nonlinear wave equation [image] modeling the dynamics of (pseudorelativistic) boson stars. For spherically symmetric initial data, u0(x) ∈ Cmath image (ℝ3), with negative energy, we prove blowup of u(t, x) in the H1/2-norm within a finite time. Physically this phenomenon describes the onset of “gravitational collapse” of a boson star. We also study blowup in external, spherically symmetric potentials, and we consider more general Hartree-type nonlinearities. As an application, we exhibit instability of ground state solitary waves at rest if m = 0.

Universality at the edge of the spectrum for unitary, orthogonal, and symplectic ensembles of random matrices

Abstract: We prove universality at the edge of the spectrum for unitary (β = 2), orthogonal (β = 1), and symplectic (β = 4) ensembles of random matrices in the scaling limit for a class of weights w(x) = e−V(x) where V is a polynomial, V(x) = κ2mx2m + · · ·, κ2m > 0. The precise statement of our results is given in Theorem 1.1 and Corollaries 1.2 and 1.4 below. For the same class of weights, a proof of universality in the bulk of the spectrum is given in [12] for the unitary ensembles and in [9] for the orthogonal and symplectic ensembles. Our starting point in the unitary case is [12], and for the orthogonal and symplectic cases we rely on our recent work [9], which in turn depends on the earlier work of Widom [46] and Tracy and Widom [42]. As in [9], the uniform Plancherel-Rotach-type asymptotics for the orthogonal polynomials found in [12] plays a central role. The formulae in [46] express the correlation kernels for β = 1, 4 as a sum of a Christoffel-Darboux (CD) term, as in the case β = 2, together with a correction term. In the bulk scaling limit [9], the correction term is of lower order and does not contribute to the limiting form of the correlation kernel. By contrast, in the edge scaling limit considered here, the CD term and the correction term contribute to the same order: this leads to additional technical difficulties over and above [49]. © 2006 Wiley Periodicals, Inc.

Almost global existence for Hamiltonian semilinear Klein‐Gordon equations with small Cauchy data on Zoll manifolds

Abstract: This paper is devoted to the proof of almost global existence results for Klein-Gordon equations on Zoll manifolds (e.g. spheres of arbitrary dimension) with Hamiltonian nonlinearities, when the Cauchy data are smooth and small. The proof relies on Birkhoff normal form methods and on the specific distribution of eigenvalues of the laplacian perturbed by a potential on Zoll manifolds.

Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity

Abstract: We study the branch of semistable and unstable solutions (i.e., those whose Morse index is at most 1) of the Dirichlet boundary value problem − Δu = λf(x)/(1 − u)2 on a bounded domain Ω ⊂ ℝN, which models—among other things—a simple electrostatic microelectromechanical system (MEMS) device. We extend the results of 11 relating to the minimal branch, by obtaining compactness along unstable branches for 1 ≤ N ≤ 7 on any domain Ω and for a large class of “permittivity profiles” f. We also show the remarkable fact that powerlike profiles f(x) ≅ |x|α can push back the critical dimension N = 7 of this problem by establishing compactness for the semistable branch on the unit ball, also for N ≥ 8 and as long as As a byproduct, we are able to follow the second branch of the bifurcation diagram and prove the existence of a second solution for λ in a natural range. In all these results, the conditions on the space dimension and on the power of the profile are essentially sharp. © 2007 Wiley Periodicals, Inc.

Action minimization and sharp‐interface limits for the stochastic Allen‐Cahn equation

Abstract: We study the action minimization problem which is formally associated to phase transformation in the stochastically perturbed Allen-Cahn equation. The sharp-interface limit is related to (but di erent from) the sharp-interface limits of the related energy functional and deterministic gradient flows. In the sharp-interface limit of the action minimization problem, we find distinct “most likely switching pathways,” depending on the relative costs of nucleation and propagation of interfaces. This competition is captured by the limit of the action functional, which we derive formally and propose as the natural candidate for the -limit. Guided by the reduced functional, we prove upper and lower bounds for the minimal action which agree on the level of scaling. (This is a preprint of an article accepted for publication in Comm. Pure App. Math, October 2005.)

Long-time asymptotics of the nonlinear Schrödinger equation shock problem

Abstract: The long-time asymptotics of two colliding plane waves governed by the focusing nonlinear Schrodinger equation are analyzed via the inverse scattering method. We find three asymptotic regions in space-time: a region with the original wave modified by a phase perturbation, a residual region with a one-phase wave, and an intermediate transition region with a modulated two-phase wave. The leading-order terms for the three regions are computed with error estimates using the steepest-descent method for Riemann-Hilbert problems. The nondecaying initial data requires a new adaptation of this method. A new breaking mechanism involving a complex conjugate pair of branch points emerging from the real axis is observed between the residual and transition regions. Also, the effect of the collision is felt in the plane-wave state well beyond the shock front at large times. © 2007 Wiley Periodicals, Inc.

Commutation relations for Schramm-Loewner evolutions

Abstract: Schramm-Loewner evolutions (SLEs) describe a one-parameter family of growth processes in the plane that have particular conformal invariance properties. For instance, SLE can define simple random curves in a simply connected domain. In this paper we are interested in questions pertaining to the definition of several SLEs in a domain (i.e., several random curves). In particular, we derive infinitesimal commutation conditions, discuss some elementary solutions, study integrability conditions following from commutation, and show how to lift these infinitesimal relations to global relations in simple cases. The situation in multiply connected domains is also discussed. © 2007 Wiley Periodicals, Inc.

On the number of interior peak solutions for a singularly perturbed Neumann problem

Abstract: We consider the following singularly perturbed Neumann problem: where Δ = Σ ∂2/∂x is the Laplace operator, ϵ > 0 is a constant, Ω is a bounded, smooth domain in ℝN with its unit outward normal ν, and f is superlinear and subcritical. A typical f is f(u) = up where 1 < p < +∞ when N = 2 and 1 < p < (N + 2)/(N − 2) when N ≥ 3. We show that there exists an ϵ0 > 0 such that for 0 < ϵ < ϵ0 and for each integer K bounded by where αN, Ω, f is a constant depending on N, Ω, and f only, there exists a solution with K interior peaks. (An explicit formula for αN, Ω, f is also given.) As a consequence, we obtain that for ϵ sufficiently small, there exists at least [αN, Ωf/ϵN (|ln ϵ|)N] number of solutions. Moreover, for each m ∈ (0, N) there exist solutions with energies in the order of ϵN−m. © 2006 Wiley Periodicals, Inc.

CMV: The unitary analogue of Jacobi matrices

Abstract: We discuss a number of properties of CMV matrices, by which we mean the class of unitary matrices studied recently by Cantero, Moral, and Velazquez. We argue that they play an equivalent role among unitary matrices to that of Jacobi matrices among all Hermitian matrices. In particular, we describe the analogues of well-known properties of Jacobi matrices: foliation by co-adjoint orbits, a natural symplectic structure, algorithmic reduction to this shape, Lax representation for an integrable lattice system (Ablowitz-Ladik), and the relation to orthogonal polynomials. As offshoots of our analysis, we will construct action/angle variables for the finite Ablowitz-Ladik hierarchy and describe the long-time behavior of this system. © 2006 Wiley Periodicals, Inc.

On a micro‐macro model for polymeric fluids near equilibrium

Abstract: In this paper, we study a micro-macro model for polymeric fluid. The system involves coupling between the macroscopic momentum equation and a microscopic evolution equation describing the combined effects of the microscopic potential and thermofluctuation. We employ an energetic variation procedure to explore the relation between the macroscopic transport of the particles and the induced elastic stress due to the microscopic structure. For the initial data not far from the equilibrium, we prove the global existence and uniqueness of classical solutions to the system. © 2006 Wiley Periodicals, Inc.

Relativistic diffusions and schwarzschild geometry

Abstract: The purpose of this article is to introduce and study a relativistic motion whose acceleration, in proper time, is given by a white noise. We deal with general relativity and consider more closely the problem of the asymptotic behavior of paths in the Schwarzschild geometry example. © 2006 Wiley Periodicals, Inc.

Global existence for three‐dimensional incompressible isotropic elastodynamics

Abstract: The existence of global-in-time classical solutions to the Cauchy problem for incompressible, nonlinear, isotropic elastodynamics for small initial displacements is proved. The generalized energy method is used to obtain strong dispersive estimates that are needed for long-time stability. This requires the use of weighted local decay estimates for the linearized equations, which are obtained as a special case of a new general result for certain isotropic symmetric hyperbolic systems. In addition, the pressure that arises as a Lagrange multiplier to enforce the incompressibility constraint is estimated as a nonlinear term. The incompressible elasticity equations are inherently linearly degenerate in the isotropic case; i.e., the equations satisfy a null condition necessary for global existence in three dimensions. c � 2007 Wiley Periodicals, Inc.

Numerical solution of the small dispersion limit of Korteweg—de Vries and Whitham equations

Abstract: The Cauchy problem for the Korteweg—de Vries (KdV) equation with small dispersion of order ϵ2, ϵ ≪ 1, is characterized by the appearance of a zone of rapid, modulated oscillations of wavelength of order ϵ. These oscillations are approximately described by the elliptic solution of KdV where the amplitude, wave number, and frequency are not constant but evolve according to the Whitham equations. In this manuscript we give a quantitative analysis of the discrepancy between the numerical solution of the KdV equation in the small dispersion limit and the corresponding approximate solution for values of ϵ between 10−1 and 10−3. The numerical results are compatible with a difference of order ϵ close to the center within the “interior” of the Whitham oscillatory zone, of order ϵ1/3 at the left boundary outside the Whitham zone and of order √ϵ at the right boundary outside the Whitham zone. © 2007 Wiley Periodicals, Inc.

Stability properties of steady water waves with vorticity

Abstract: We present two stability analyses for exact periodic traveling water waves with vorticity. The first approach leads in particular to linear stability properties of water waves for which the vorticity decreases with depth. The second approach leads to a formal stability property for long water waves that have small vorticity and amplitude although we do not use a small-amplitude or long-wave approximation. © 2006 Wiley Periodicals, Inc.

Velocity averaging, kinetic formulations, and regularizing effects in quasi‐linear PDEs

Abstract: We prove in this paper new velocity-averaging results for second-order multidimensional equations of the general form L(∇x ,v )f (x ,v ) = g(x ,v )where L(∇x ,v ):= a(v) ·∇ x −∇ � ·b(v)∇x. These results quantify the Sobolev regularity of the averages, � v f (x, v)φ(v)dv, in terms of the nondegeneracy of the set

Decay estimates for the wave and Dirac equations with a magnetic potential

Abstract: We study the electromagnetic wave equation and the perturbed massless Dirac equation on R t × R 3 : u tt - (∇ + iA(x)) 2 u + B(x) u = 0, iu t - Du + V(x)u = 0, where the potentials A(x), B(x), and V(x) are assumed to be small but may be rough. For both equations, we prove the expected time decay rate of the solution |u(t, x)| ≤ 1 t ∥f∥X where the norm ∥f∥ x can be expressed as the weighted L 2 -norm of a few derivatives of the data f.

Probing for electrical inclusions with complex spherical waves

Abstract: Let a physical body Ω in ℝ2 or ℝ3 be given. Assume that the electric conductivity distribution inside Ω consists of conductive inclusions in a known smooth background. Further, assume that a subset Γ ⊂ ∂Ω is available for boundary measurements. It is proved using hyperbolic geometry that certain information about the location of the inclusions can be exactly recovered from static electric measurements on Γ. More precisely: given a ball B with center outside the convex hull of Ω and satisfying (B ∩ ∂Ω) ⊂ Γ, boundary measurements on Γ with explicitly given Dirichlet data are enough to determine whether B intersects the inclusion. An approximate detection algorithm is introduced based on the theory. Numerical experiments in dimension two with simulated noisy data suggest that the algorithm finds the inclusion-free domain near Γ and is robust against measurement noise. © 2007 Wiley Periodicals, Inc.

A constant rank theorem for solutions of fully nonlinear elliptic equations

Abstract: Convexity is an important geometric property associated with the study of partial differential equations, in particular for equations related to problems in differential geometry. There is a vast literature on this subject. In an important development in 1985, a technique was devised to deal with the convexity issue via the homotopy method of deformation in the work of Caffarelli and Friedman [7]. In [7], the existence of convex solutions for semilinear elliptic equations in two dimensions was proved by a form of deformation lemma using the strong maximum principle (see also the work of Singer, Wong, Yau, and Yau [17] for a similar approach). The core of this approach is the establishment of the constant rank theorem; that is, the rank of the Hessian of the corresponding convex solution is constant. The result in [7] was later generalized to higher dimensions in [15]. The constant rank theorem is a refined statement of convexity. This has profound implications in the geometry of solutions. The idea of the deformation lemma and the establishment of the constant rank theorem can be extended to various nonlinear differential equations in differential geometry involving symmetric curvature tensors. Recently, in connection to the Christoffel-Minkowski problem and the problem of prescribing Weingarten curvatures in classical differential geometry, this form of deformation lemma was extended to some equations involving the second fundamental forms of embedded hypersurfaces in R [11, 12, 13]. The constant rank theorem shares similar geometric flavors in spirit with a classical theorem of Hartman and Nirenberg [14], where they treated hypersurfaces in R with a vanishing spherical Jacobian. A pertinent question is under what structural conditions for partial differential equations is the positivity of the symmetric curvature tensor preserved under homotopy deformation? The purpose of this paper is to establish a general principle

Stability and convergence of efficient Navier‐Stokes solvers via a commutator estimate

Abstract: For strong solutions of the incompressible Navier-Stokes equations in bounded domains with velocity specified at the boundary, we establish the unconditional stability and convergence of discretization schemes that decouple the updates of pressure and velocity through explicit time stepping for pressure. These schemes require no solution of stationary Stokes systems, nor any compatibility between velocity and pressure spaces to ensure an inf-sup condition, and are representative of a class of highly efficient computational methods that have recently emerged. The proofs are simple, based upon a new, sharp estimate for the commutator of the Laplacian and Helmholtz projection operators. This allows us to treat an unconstrained formulation of the Navier-Stokes equations as a perturbed diffusion equation. c � 2006 Wiley Periodicals, Inc.

The mean‐field approximation in quantum electrodynamics: The no‐photon case

Abstract: We study the mean-field approximation of quantum electrodynamics (QED) by means of a thermodynamic limit. The QED Hamiltonian is written in Coulomb gauge and does not contain any normal ordering or choice of bare electron/positron subspaces. Neglecting photons, we properly define this Hamiltonian in a finite box [−L/2; L/2)3, with periodic boundary conditions and an ultraviolet cutoff λ. We then study the limit of the ground state (i.e., the vacuum) energy and of the minimizers as L goes to infinity, in the Hartree-Fock approximation. In the case with no external field, we prove that the energy per volume converges and obtain in the limit a translation-invariant projector describing the free Hartree-Fock vacuum. We also define the energy per unit volume of translation-invariant states and prove that the free vacuum is the unique minimizer of this energy. In the presence of an external field, we prove that the difference between the minimum energy and the energy of the free vacuum converges as L goes to infinity. We obtain in the limit the so-called Bogoliubov-Dirac-Fock functional. The Hartree-Fock (polarized) vacuum is a Hilbert-Schmidt perturbation of the free vacuum and it minimizes the Bogoliubov-Dirac-Fock energy. © 2006 Wiley Periodicals, Inc.