Showing papers in "Communications on Pure and Applied Mathematics in 1994"

Hyperbolic conservation laws with stiff relaxation terms and entropy

Abstract: We study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms. Reduced systems, inviscid and viscous local conservation laws, and weakly nonlinear limits are derived through asymptotic expansions. An entropy condition is introduced for N × N systems that ensures the hyperbolicity of the reduced inviscid system. The resulting characteristic speeds are shown to be interlaced with those of the original system. Moreover, the first correction to the reduced system is shown to be dissipative. A partial converse is proved for 2 × 2 systems. This structure is then applied to study the convergence to the reduced dynamics for the 2 × 2 case. © 1994 John Wiley & Sons, Inc.

The τ‐function of the universal whitham hierarchy, matrix models and topological field theories

Abstract: The universal Witham hierarchy is considered from the point of view of topological field theories. The �-function for this hierarchy is defined. It is proved that the algebraic orbits of Whitham hierarchy can be identified with various topological matter models coupled with topological gravity.

On the maslov index

Abstract: In this paper we give four definitions of Maslov index and show that they all satisfy the same system of axioms and hence are equivalent to each other. Moreover, relationships of several symplectic and differential geometric, analytic, and topological invariants (including triple Maslov indices, eta invariants, spectral flow and signatures of quadratic forms) to the Maslov index are developed and formulae relating them are given. The broad presentation is designed with a view to applications both in geometry and in analysis. © 1994 John Wiley & Sons, Inc.

Renormalization Group and Asymptotics of Solutions of Nonlinear Parabolic Equations

Abstract: We present a general method for studying long-time asymptotics of nonlinear parabolic partial differential equations. The method does not rely on a priori estimates such as the maximum principle. It applies to systems of coupled equations, to boundary conditions at infinity creating a front, and to higher (possibly fractional) differential linear terms. We present in detail the analysis for nonlinear diffusion-type equations with initial data falling off at infinity and also for data interpolating between two different stationary solutions at infinity. In an accompanying paper, [5], the method is applied to systems of equations where some variables are ''slaved,'' such as the complex Ginzburg-Landau equation. (c) 1994 John Wiley & Sons, Inc.

The collisionless shock region for the long-time behavior of solutions of the KdV equation

Abstract: The authors further develop the nonlinear steepest descent method of [5] and [6] to give a full description of the collisionless shock region for solutions of the KdV equation with decaying initial data. © 1994 John Wiley & Sons, Inc.

Potential theory for regular and mach reflection of a shock at a wedge

Abstract: If a plane shock hits a wedge, a self-similar pattern of reflected shocks travels outward as the shock moves forward in time. The nature of the pattern is explored for weak incident shocks (strength b) and small wedge angles 2θw through potential theory, a number of different scalings, some study of mixed equations and matching asymptotics for the different scalings. The self-similar equations are of mixed type. A linearization gives a linear mixed flow valid away from a sonic curve. Near the sonic curve a shock solution is constructed in another scaling except near the zone of interaction between the incident shock and the wall where a special scaling is used. The parameter β = c1θ2w(γ + 1)b ranges from 0 to ∞. Here γ is the polytropic constant and C1 is the sound speed behind the incident shock. For β > 2 regular reflection (weak or strong) can occur and the whole pattern is reconstructed to lowest order in shock strength. For β < 1/2 Mach reflection occurs and the flow behind the reflection is subsonic and can be constructed in principle (with an open elliptic problem) and matched. The case β = 0 can be solved. For 1/2 < β < 2 or even larger β the flow behind a Mach reflection may be transonic and further investigation must be made to determine what happens. The basic pattern of reflection is an almost semi-circular shock issuing, for regular reflection, from the reflection point on the wedge and for Mach reflection, matched with a local interaction flow. Assuming their nature, choosing the least entropy generation, the weak regular reflection will occur for β sufficiently large (von Neumann paradox). An accumulation point of vorticity occurs on the wedge above the leading point. © 1994 John Wiley & Sons, Inc.

Asymptotics of heavy atoms in high magnetic fields: I. Lowest landau band regions

Abstract: The ground state energy of an atom of nuclear charge Ze in a magnetic field B is evaluated exactly to leading order as Z ∞. In this and a companion work (see [28]) we show that there are five regions as Z ∞: B Z3. Regions 1, 2, 3, and 4 (and conceivably 5) are relevant for neutron stars. Different regions have different physics and different asymptotic theories. Regions 1, 2, and 3 are described by a simple density functional theory of the semiclassical Thomas-Fermi form. Here we concentrate mainly on regions 4 and 5 which cannot be so described, although 3, 4, and 5 have the common feature (as shown here) that essentially all electrons are in the lowest Landau band. Region 5 does have, however, a simple non-classical density functional theory (which can be solved exactly). Region 4 does not, but, surprisingly, it can be described by a novel density matrix functional theory. © 1994 John Wiley & Sons, Inc.

L1 asymptotic behavior of compressible, isentropic, viscous 1-D flow

Abstract: We study the large time behavior in L1 of the compressible, isentropic, viscous 1-D flow. Under the assumption that the initial data are smooth and small, we show that the solutions are approximated by the solutions of a parabolic system, and in turn by diffusion waves, which are solutions of Burgers equations. Decay rates in L1 are obtained. Our method is based on the study of pointwise properties in the physical space of the fundamental solution to the linearized system. © 1994 John Wiley & Sons, Inc.

Existence and positivity of solutions of a fourth‐order nonlinear PDE describing interface fluctuations

Abstract: We study the partial differential equation which arose originally as a scaling limit in the study of interface fluctuations in a certain spin system. In that application x lies in R, but here we study primarily the periodic case x E S'. We establish existence, uniqueness, and regularity of solutions, locally in time, for positive initial data in H' (S' ), and prove the existence of several families of Lyapunov functions for the evolution. From the latter we establish a sharp connection between existence globally in time and positivity preservation: if [O,T*) is a maximal half open interval of existence for a positive solution of the equation, with T' (1 + &)/16r2& then T* = 03 and lim,-m w(r, .) exists and is constant. We discuss also some explicit solutions and propose a generalization to higher dimensions. 0 1994 John Wiley &

Existence of global weak solutions to one-component vlasov-poisson and fokker-planck-poisson systems in one space dimension with measures as initial data

Abstract: We consider Cauchy problems for the 1-D one component Vlasov-Poisson and Fokker-Planck-Poisson equations with the initial electron density being in the natural space of arbitrary non-negative finite measures. In particular, the initial density can be a Dirac measure concentrated on a curve, which we refer to as “electron sheet” initial data. These problems resemble both structurally and functional analytically Cauchy problems for the 2-D Euler and Navier-Stokes equations (in vorticity formulation) with vortex sheet initial data. Here, we need to define weak solutions more specifically than usual since the product of a finite measure with a function of bounded variation is involved. We give a natural definition of the product, establish its weak stability, and existence of weak solutions follows. Our concept of weak solutions through the newly defined product is justified since solutions to the Fokker-Planck-Poisson equation, the analogue of Navier-Stokes equation, are shown to converge to weak solutions of the Vlasov-Poisson equation as the Fokker-Planck term vanishes. The main difficulty is the aforementioned weak stability which we establish through a careful analysis of the explicit structure of these equations. This is needed because the problem studied here is beyond the range of applicability of the “velocity averaging” compactness methods of DiPerna-Lions. © 1994 John Wiley & Sons, Inc.

Shape theorem, lyapounov exponents, and large deviations for brownian motion in a poissonian potential

Abstract: We derive a large deviation principle governing the position of a d-dimensional Brownian motion moving in a Poissonian potential. The derivation of this large deviation principle, and the form of the rate function rely on a result similar to the “shape theorem” of first passage percolation. This result produces certain constants which play in this multidimensional situation a similar role as the Lyapounov exponents in the one-dimensional case. The large deviation principle enables us to investigate the transition of regime, which occurs between the small ∣h∣ and the large ∣h∣ case, for Brownian motion with a constant drift h moving in the same potential. © 1994 John Wiley & Sons, Inc.

Hydrodynamical limit for a nongradient system: The generalized symmetric exclusion process

Abstract: We consider a symmetric simple exclusion process where at most two particles per site are permitted This model turns out to be nongradient We prove that the particles' densities, under a diffusive rescaling of space and time, converge to the solution of a diffusion equation We give a variational characterization of the diffusion coefficent We also prove, for the generator of the process in finite volume, a lower bound on the spectral gap uniform in the volume © 1994 John Wiley & Sons, Inc

On Global Representation of Lagrangian Distributions and Solutions of Hyperbolic Equations

Abstract: In this paper we develop a new approach to the theory of Fourier integral operators. It allows us to represent the Schwartz kernel of a Fourier integral operator by one oscillatory integral with a complex phase function. We consider Fourier integral operators associated with canonical transformations, having in mind applications to hyperbolic equations. As a by-product we obtain yet another formula for the Maslov index. (C) 1994 John Wiley & Sons, Inc.

Pointwise fourier inversion and related eigenfunction expansions

Abstract: Necessary and sufficient conditions are found for the convergence at a pre-assigned point of the spherical partial sums (resp. integrals) of the Fourier series (resp. integral) in the class of piecewise smooth functions on Euclidean space. These results carry over unchanged to spherical harmonic expansions, Fourier transforms on hyperbolic space, and Dirichlet eigenfunction expansions with respect to the Laplace operator on a class of Riemannian manifolds. © 1994 John Wiley & Sons, Inc.

On the numerical solution of two-point boundary value problems II

Abstract: : In a recent paper Greengard and Rokhlin introduce a numerical technique for the rapid solution of integral equations resulting from linear two-point boundary value problems for second order ordinary differential equations. In this paper, we extend the method to systems of ordinary differential equations. After reducing the system of differential equations to a system of second kind integral equations, we discretize the latter via a high order Nystrom scheme. A somewhat involved analytical apparatus is then constructed which allows for the solution of the discrete system using O(N . p squared . n cubed) operations, with N the number of nodes on the interval, p the desired order of convergence, and n the number of equations in the system. Thus, the advantages of the integral equation formulation (small condition number, insensitivity to boundary layers, insensitivity to end-point singularities, etc. ) are retained, while achieving a computational efficiency previously available only to finite difference of finite element methods. We in addition present a Newton method for solving boundary value problems for nonlinear first order systems in which each Newton iterate is the solution of a second kind integral equation; the analytical and numerical advantages of integral equations are thus obtained for nonlinear boundary value problems. (kr)

Morse‐type information on palais‐smale sequences obtained by min‐max principles

Abstract: In the context of the min-max approach to critical point theory, but without the usual compactness assumptions a la Palais-Smale and the nondegeneracy conditions a la Fredholm, we construct almost critical points with two-sided estimates on their approximate Morse indices which are arbitrarily close to certain—a priori given—dual sets. This additional topological and analytical information about almost critical sequences can sometimes be crucial in the proof of their convergence and therefore in solving the corresponding variational problem. © 1994 John Wiley & Sons, Inc.

On the nonlinear stability of plane waves for the ginzburg‐landau equation

Abstract: I consider the nonlinear stability of plane wave solutions to a Ginzburg-Landau equation with additional fifth-order terms and cubic terms containing spatial derivatives. I show that, under the constraint that the diffusion coefficient be real, these waves are stable. Furthermore, it is shown that the radial component of the perturbation decays at a faster rate than the phase component of the perturbation as t ∞. The result is also applicable to the classical Ginzburg-Landau equation. © 1994 John Wiley & Sons, Inc.