Showing papers in "Communications on Pure and Applied Mathematics in 1999"

Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory

Abstract: We consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn(x)dx = e−nV(x)dx on the line as n ∞. The potentials V are assumed to be real analytic, with sufficient growth at infinity. The principle results concern Plancherel-Rotach-type asymptotics for the orthogonal polynomials down to the axis. Using these asymptotics, we then prove universality for a variety of statistical quantities arising in the theory of random matrix models, some of which have been considered recently in [31] and also in [4]. An additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials (see also [4]). The orthogonal polynomial problem is formulated as a Riemann-Hilbert problem following [19, 20]. The Riemann-Hilbert problem is analyzed in turn using the steepest-descent method introduced in [12] and further developed in [11, 13]. A critical role in our method is played by the equilibrium measure dμV for V as analyzed in [8]. © 1999 John Wiley & Sons, Inc.

Strong asymptotics of orthogonal polynomials with respect to exponential weights

Abstract: We consider asymptotics of orthogonal polynomials with respect to weights w(x)dx= e Q(x) dx on the real line, where Q(x)=∑ 2m k=0 qkx k , q2m> 0, denotes a polynomial of even order with positive leading coefficient. The orthogonal polynomial problem is formulated as a Riemann-Hilbert problem following [22, 23]. We employ the steepest-descent-type method introduced in [18] and further developed in [17, 19] in order to obtain uniform Plancherel-Rotach-type asymptotics in the entire complex plane, as well as asymptotic formulae for the zeros, the leading coefficients, and the recurrence coefficients of the orthogonal polynomials. c 1999 John Wiley & Sons, Inc.

Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations

Abstract: The three-dimensional motion of an incompressible inviscid fluid is classically described by the Euler equations but can also be seen, following Arnold [1], as a geodesic on a group of volume-preserving maps. Local existence and uniqueness of minimal geodesics have been established by Ebin and Marsden [16]. In the large, for a large class of data, the existence of minimal geodesics may fail, as shown by Shnirelman [26]. For such data, we show that the limits of approximate solutions are solutions of a suitable extension of the Euler equations or, equivalently, are sharp measure-valued solutions to the Euler equations in the sense of DiPerna and Majda [14]. © 1999 John Wiley & Sons, Inc.

FETI and Neumann--Neumann Iterative Substructuring Methods: Connections and New Results

Abstract: The FETI and Neumann-Neumann families of algorithms are among the best know and most severely tested domain decomposition methods for elliptic partial differential equations. They are iterative substructuring methods and have many algorithmic components in common but there are also differences. The purpose of this paper is to further unify the theory for these two families of methods and to introduce a new family of FETI algorithms. Bounds on the rate of convergence, which are uniform with respect to the coefficients of a family of elliptic problems with heterogeneous coefficients, are established for these new algorithms. The theory for a variant of the Neumann--Neumann algorithm is also redeveloped stressing similarities to that for the FETI methods.

The Semiclassical Limit of the Defocusing NLS Hierarchy

Abstract: We establish the semiclassical limit of the one-dimensional defocusing cubic nonlinear Schrodinger (NLS) equation. Complete integrability is exploited to obtain a global characterization of the weak limits of the entire NLS hierarchy of conserved densities as the field evolves from reflectionless initial data under all the associated commuting flows. Consequently, this also establishes the zero-dispersion limit of the modified Korteweg‐de Vries equation that resides in that hierarchy. We have adapted and clarified the strategy introduced by Lax and Levermore to study the zero-dispersion limit of the Korteweg‐de Vries equation, expanding it to treat entire integrable hierarchies and strengthening the limits obtained. A crucial role is played by the convexity of the underlying log-determinant with respect to the times associated with the commuting flows. c 1999 John Wiley & Sons, Inc.

Entire solutions of the KPP equation

Abstract: This paper deals with the solutions defined for all time of the KPP equation ut = uxx+ f(u); 0 0, f 0 (1) 0i n(0; 1), and f 0 (s) f 0 (0) in [0; 1]. This equation admits infinitely many traveling-wave-type solutions, increasing or decreasing in x .I t also admits solutions that depend only on t. In this paper, we build four other manifolds of solutions: One is 5-dimensional, one is 4-dimensional, and two are 3-dimensional. Some of these new solutions are obtained by considering two traveling waves that come from both sides of the real axis and mix. Furthermore, the traveling-wave solutions are on the boundary of these four manifolds. c 1999 John Wiley & Sons, Inc.

A regularity theory of biharmonic maps

Abstract: In this article we prove the regularity of weakly biharmonic maps of domains in Euclidean four space into spheres, as well as the corresponding partial regu- larity result of stationary biharmonic maps of higher-dimensional domains into spheres. c 1999 John Wiley & Sons, Inc. In this article we consider for simplicity the class of bihar monic maps from Eu- clidean domains to spheres. We realize the standard spheres S k as unit vectors in R k+1 , and consider maps u : Ω ! S k as vector-valued functions that are contained in S k . The energy functional for biharmonic maps is then Ω j Δuj 2 dx. A locally de- fined biharmonic map is a map that is critical with respect to c ompactly supported variations. We note that in the case where the domain has dimension four, this en- ergy functional is conformally invariant, and hence conformal maps of Euclidean four-space are biharmonic in this sense. We remark that this definition of bihar- monic map depends on the embedding of the target space in Euclidean space. We

Well‐posedness theory for hyperbolic conservation laws

Abstract: The paper presents a well-posedness theory for the initial value problem for a general system of hyperbolic conservation laws. We will start with the refinement of Glimm's existence theory and discuss the principle of nonlinear through wave tracing. Our main goal is to introduce a nonlinear functional for two solutions with the property that it is equivalent to the L1(x) distance between the two solutions and is time-decreasing. Moreover, the functional is constructed explicitly in terms of the wave patterns of the solutions through the nonlinear superposition. It consists of a linear term measuring the L1(x) distance, a quadratic term measuring the coupling of waves and distance, and a generalized entropy functional. © 1999 John Wiley & Sons, Inc.

Aubry-mather theory and periodic solutions of the forced burgers equation

Abstract: Consider a Hamiltonian system with Hamiltonian of the form H(x;t;p) where H is convex in p and periodic in x ,a ndt and x2 R 1 . It is well-known that its smooth invariant curves correspond to smoothZ 2 -periodic solutions of the PDE ut +H(x;t;u)x =0 : In this paper, we establish a connection between the Aubry-Mather theory of invariant sets of the Hamiltonian system andZ 2 -periodic weak solutions of this PDE by realizing the Aubry-Mather sets as closed subsets of the graphs of these weak solutions. We show that the complement of the Aubry-Mather set on the graph can be viewed as a subset of the generalized unstable manifold of the Aubry-Mather set, defined in (2.24). The graph itself is a backward-invariant set of the Hamiltonian system. The basic idea is to embed the globally minimizing orbits used in the Aubry-Mather theory into the characteristic fields of the above PDE. This is done by making use of one- and two-sided minimizers, a notion introduced in [12] and inspired by the work of Morse on geodesics of type A [26]. The asymptotic slope of the minimizers, also known as the rotation number, is given by the derivative of the homogenized Hamiltonian, defined in [21]. As an application, we prove that theZ 2 -periodic weak solution of the above PDE with given irrational asymptotic slope is unique. A similar connection also exists in multidimensional problems with the convex Hamiltonian, except that in higher dimensions, two-sided minimizers with a specified asymptotic slope may not exist. c 1999 John Wiley & Sons, Inc.

Zero-viscosity limit of the linearized Navier-Stokes equations for a compressible viscous fluid in the half-plane

Abstract: The zero-viscosity limit for an initial boundary value problem of the linearized Navier-Stokes equations of a compressible viscous fluid in the half-plane is studied. By means of the asymptotic analysis with multiple scales, we first construct an approximate solution of the linearized problem of the Navier-Stokes equations as the combination of inner and boundary expansions. Next, by carefully using the technique on energy methods, we show the pointwise estimates of the error term of the approximate solution, which readily yield the uniform stability result for the linearized Navier-Stokes solution in the zero-viscosity limit. © 1999 John Wiley & Sons, Inc.

Evolution of microstructure in unstable porous media flow: A relaxational approach

Abstract: We study the flow of two immiscible fluids of different density and mobility in a porous medium. If the heavier phase lies above the lighter one, the interface is observed to be unstable. The two phases start to mix on a mesoscopic scale and the mixing zone grows in time—an example of evolution of microstructure. A simple set of assumptions on the physics of this two-phase flow in a porous medium leads to a mathematically ill-posed problem—when used to establish a continuum free boundary problem. We propose and motivate a relaxation of this “nonconvex” constraint of a phase distribution with a sharp interface on a macroscopic scale. We prove that this approach leads to a mathematically well-posed problem that predicts shape and evolution of the mixing profile as a function of the density difference and mobility quotient. © 1999 John Wiley & Sons, Inc.

Asymptotic and limiting profiles of blowup solutions of the nonlinear schrodinger equation with critical power

Abstract: This paper is a sequel to previous ones 38, 39, 41. We continue the study of the blowup problem for the nonlinear Schrodinger equation with critical power nonlinearity (NSC). We introduce a new idea to prove the existence of a blowup solution in H1(ℝN) without any weight condition and reduce the problem to a kind of variational problem. Our new method refines the previous results concerning the asymptotic and limiting profiles of blowup solutions: For a certain class of initial data, the blowup solution behaves like a finite superposition of zero-energy, H1-bounded, global-in-time solutions of (NSC); these singularities stay in a bounded region in ℝN, and one can see that the so-called shoulder emerges outside these singularities as suggested by some numerical computations (see, e.g., [26]). We investigate the asymptotic behavior of zero-energy, global-in-time solutions of (NSC) and find that such a solution behaves like a “multisoliton.” However, it is not an assemblage of free “particles”; the “solitons” interact with each other. © 1999 John Wiley & Sons, Inc.

Statistical equilibrium measures and coherent states in two-dimensional turbulence

Abstract: The equilibrium statistics of the Euler equations in two dimensions are studied, and a new continuum model of coherent, or organized, states is proposed. This model is defined by a maximum entropy principle similar to that governing the Miller-Robert model except that the family of global vorticity invariants is relaxed to a family of inequalities on all convex enstrophy integrals. This relaxation is justified by constructing the continuum model from a sequence of lattice models defined by Gibbs measures whose invariants are derived from the exact vorticity dynamics, not a spectral truncation or spatial discretization of it. The key idea is that the enstrophy integrals can be partially lost to vorticity fluctuations on a range of scales smaller than the lattice microscale, while energy is retained in the larger scales. A consequence of this relaxation is that many of the convex enstrophy constraints can be inactive in equilibrium, leading to a simplification of the mean-field equation for the coherent state. Specific examples of these simplified theories are established for vortex patch dynamics. In particular, a universal relation between mean vorticity and stream function is obtained in the dilute limit of the vortex patch theory, which is different from the sinh relation predicted by the Montgomery-Joyce theory of point vortices. © 1999 John Wiley & Sons, Inc.

Direct computation of multivalued phase space solutions for Hamilton-Jacobi equations

Abstract: A method is presented for the exact and complete resolution of Hamiltonian systems only using the corresponding Hamilton-Jacobi equation (we show that it possible to get the ODE solution from the corresponding PDE).

Persistence of Overflowing Manifolds for Semiflow

Abstract: This paper, which is a sequel to a previous one [4] by the same authors, is devoted to the persistence of overflowing manifolds and inflowing manifolds for a semiflow in a Banach space. We consider a C1 semiflow defined on a Banach space X; that is, it is continuous on [0,∞)×X , and for each t ≥ 0, T t : X → X is C1, and T t ◦ T (x) = T t+s(x) for all t, s ≥ 0 and x ∈ X . A typical example is the solution operator for a differential equation. In [4] we proved that a compact, normally hyperbolic, invariant manifold M persists under small C1 perturbations in the semiflow. We also showed that in a neighborhood of M , there exist a center-stable manifold and a center-unstable manifold that intersect in the manifold M . In [4] the compactness and invariance of the manifold M were important assumptions. In the present paper, we study the more general case where the manifold M is overflowing (“negatively invariant and the semiflow crosses the boundary transversally”) or inflowing (“positively invariant and the semiflow crosses the boundary transversally”). We do not assume that M is compact or finite-dimensional. Also, M is not necessarily an imbedded manifold, but an immersed manifold. As an example, a local unstable manifold of an equilibrium point is an overflowing manifold. In brief, our main results on the overflowing manifolds may be summarized as follows (the precise statements are given in Section 2). We assume that the immersed manifoldM does not twist very much locally,M is covered by the image under T t of a subset a positive distance away from boundary, DT t contracts along the normal direction and does so more strongly than it does along the tangential direction, and DT t has a certain uniform continuity in a neighborhood of M . If the C1 perturbation T t of T t is sufficiently close to T t, then T t has a unique C1 immersed overflowing manifold M nearM . Furthermore, if T t isCk and a spectral gap condition holds, then M isCk. Similar results for inflowing manifolds are also obtained and given in Section 7.

Existence and modulation of traveling waves in particles chains

Abstract: We consider an infinite particle chain whose dynamics are governed by the following system of differential equations: where qn(t) is the displacement of the nth particle at time t along the chain axis and denotes differentiation with respect to time. We assume that all particles have unit mass and that the interaction potential V between adjacent particles is a convex C∞ function. For this system, we prove the existence of C∞, time-periodic, traveling-wave solutions of the form qn(t) = q(wt kn + where q is a periodic function q(z) = q(z+1) (the period is normalized to equal 1), ω and k are, respectively, the frequency and the wave number, is the mean particle spacing, and can be chosen to be an arbitrary parameter. We present two proofs, one based on a variational principle and the other on topological methods, in particular degree theory. For small-amplitude waves, based on perturbation techniques, we describe the form of the traveling waves, and we derive the weakly nonlinear dispersion relation. For the fully nonlinear case, when the amplitude of the waves is high, we use numerical methods to compute the traveling-wave solution and the non-linear dispersion relation. We finally apply Whitham's method of averaged Lagrangian to derive the modulation equations for the wave parameters α, β, k, and ω. © 1999 John Wiley & Sons, Inc.

Noncommutative and commutative integrability of generic toda flows in simple lie algebras

Abstract: 1.1. The Toda lattice equation introduced by Toda as a Hamilton equation describing the motion of the system of particles on the line with an exponential interaction between closest neighbours gave rise to numerous important generalizations and helped to discover many of the exciting phenomena in the theory of integrable equations. In Flaschka’s variables [F] the finite non-periodic Toda lattice describes an isospectral evolution on the set of tri-diagonal matrices in sl(n). It was explicitly solved and shown to be completely integrable by Moser [Mo] . In his paper [K1], Kostant comprehensively studied the generalization of Toda lattice that evolves on the set of “tri-diagonal” elements of a semisimple Lie algebra g which also turned out to be completely integrable with Poisson commuting integrals being provided by the Chevalley invariants of the algebra. Moreover, in this paper, as well as in the works by Ol’shanetsky and Perelomov [OP], Reyman and Semenov-Tian-Shansky [RSTS1], Symes [Sy], the method of the explicit integration of the Toda equations was extended to the case when evolution takes place on the dual space of the Borel subalgebra of g . This space is foliated into symplectic leaves of different dimensions and the natural question is what can be said about the Liouville complete integrability of the Toda flows on each of these leaves. In the particular case of generic symplectic leaves in sl(n) the complete integrability was proved by Deift, Li, Nanda and Tomei [DLNT]. This paper was motivated by the work [DLNT] and its Lie algebraic interpretation proposed in [S1], [S2], [EFS]. Our main result is the following

How rare are multiple eigenvalues

Abstract: Let A(q) be a differentiable family of self-adjoint operators on a Hilbert space H, indexed by a parameter q that belongs to a separable Banach manifold X. Assume that the spectrum of each operator A(q) is discrete, of finite multiplicity, and with no finite accumulation points. We introduce a new concept of codimension in infinite-dimensional space and then prove that under an appropriate transversality condition, related to the strong Arnold hypothesis, the members of the family A(q) having multiple eigenvalues form a set of codimension at least 2. Using this, we show that a generic member of the family A(q) has a simple spectrum (i.e., no repeated eigenvalues) and that any two values q1 and q2 of the parameter can be connected by an analytic curve γ in X such that A(q) has a simple spectrum for all q in the interior of γ. We then apply these results in two cases of physical interest: to the Laplace operator with the domain as parameter and to the Schrodinger operator with a symmetric potential as parameter. © 1999 John Wiley & Sons, Inc.

A characterization of the tight 3‐sphere II

Abstract: Recall that every closed, oriented, connected 3-manifold M admits a contact form λ. We shall give conditions on a periodic orbit P0 of the Reeb vector field defined by λ that allows the construction of an open book decomposition of M \ P0 into embedded planes asymptotic to P0 such that P0 is the binding orbit of the decomposition. It follows that M is the tight 3-sphere. As a by-product, a new dynamical criterion for the tightness of a contact structure is derived in the proof. The results extend earlier results. © 1999 John Wiley & Sons, Inc.

Stability of Cahn-Hilliard fronts

Abstract: We prove stability of the kink solution of the Cahn-Hilliard equation partial derivative(t)u = partial derivative(x)(2)(-partial derivative(x)(2)u - u/2 + u(3)/2), x is an element of R. The proof is based on an inductive renormalization group method, and we obtain detailed asymptotics of the solution as t --> infinity. We prove stability of the kink solution of the Cahn-Hilliard equation partial derivative(t)u = partial derivative(x)(2)(-partial derivative(x)(2)u - u/2 + u(3)/2), x is an element of R. The proof is based on an inductive renormalization group method, and we obtain detailed asymptotics of the solution as t --> infinity. (C) 1999 John Wiley & Sons, Inc.

Unresolved Computation and Optimal Predictions

Abstract: We present methods for predicting the solution of time-dependent partial differential equations when that solution is so complex that it cannot be properly resolved numerically, but when prior statistical information can be found. The sparse numerical data are viewed as constraints on the solution, and the gist of our proposal is a set of methods for advancing the constraints in time so that regression methods can be used to reconstruct the mean future. For linear equations we offer general recipes for advancing the constraints; the methods are generalized to certain classes of nonlinear problems, and the conditions under which strongly nonlinear problems and partial statistical information can be handled are briefly discussed. Our methods are related to certain data acquisition schemes in oceanography and meteorology. c 1999 John Wiley & Sons, Inc.

Pointwise Green's Function Approach to Stability for Scalar Conservation Laws

Abstract: We study the pointwise behavior of perturbations from a viscous shock solution to a scalar conservation law, obtaining an estimate independent of shock strength. We find that for a perturbation with initial data decaying algebraically or slower, the perturbation decays in time at the rate of decay of the integrated initial data in any L p norm, p 1. Stability in any L p norm is a direct consequence. The approach taken is that of obtaining pointwise estimates on the perturbation through a Duhamel’s principle argument that employs recently developed pointwise estimates on the Green’s function for the linearized equation. c 1999 John Wiley & Sons, Inc.

A new entropy functional for a scalar conservation law

Abstract: In this paper we introduce a new entropy functional for a scalar convex conservation law that generalizes the traditional concept of entropy of the second law of thermodynamics. The generalization has two aspects: The new entropy functional is defined not for one but for two solutions. It is defined in terms of the L1 distance between the two solutions as well as the variations of each separate solution. In addition, it is decreasing in time even when the solutions contain no shocks and is therefore stronger than the traditional entropy even in the case when one of the solutions is zero. © 1999 John Wiley & Sons, Inc.

Uniqueness for critical nonlinear wave equations and wave maps via the energy inequality

Abstract: We show uniqueness of sufficiently regular solutions to critical semilinear wave equations and wave maps in the (a priori) much larger class of distribution solutions with finite energy, assuming only that the energy is nonincreasing in time. © 1999 John Wiley & Sons, Inc.