Showing papers in "Communications on Pure and Applied Mathematics in 1985"

Asymptotically self‐similar blow‐up of semilinear heat equations

Abstract: : This reprint studies the blow-up of solutions of a nonlinear heat equation. The main goal is to characterize the asymptotic behavior of u near a singularity, assuming a suitable upper bound on the rate of blowup. Additional keywords: Liouville theorem; and Asymptotic normality.

Uniform decay estimates and the lorentz invariance of the classical wave equation

Abstract: On etudie le comportement asymptotique des solutions de □u=0 ou □=∂ t 2 −∂ 1 2 ...−∂ n 2 pour des conditions initiales u=0, u t =g(x) en t=0, avec g reguliere a support compact dans R n

Normal forms and quadratic nonlinear Klein‐Gordon equations

Abstract: On construit une transformation qui change l'equation a non-linearite cubique que l'on sait resoudre a (3+1) dimensions. On demontre que la solution du probleme original a le meme comportement asymptotique que l'equation de Klein-Gordon non lineaire

Determining conductivity by boundary measurements II. Interior results

Abstract: : In a recent paper the authors showed that an unknown real-analytic conductivity gamma may be determined from static boundary measurements. In this document they extend this analysis by demonstrating that a similar result holds for piecewise real-analytic conductivities. In addition, for the special case of a layered structure it is shown that a three times continuously differentiable conductivity is identifiable by boundary measurements. Originator-supplied keywords include: Inversion, Convergence, Algorithms, Estimates, Coefficients.

The dirichlet problem for nonlinear second‐order elliptic equations. II. Complex monge‐ampère, and uniformaly elliptic, equations

Abstract: On considere le probleme de Dirichlet pour des equations elliptiques non lineaires d'ordre 2 pour une fonction reelle dans un domaine borne Ω de R n a frontiere lisse ∂Ω

Global existence of small amplitude solutions to nonlinear klein-gordon equations in four space-time dimensions

Abstract: Consider the following class of nonlinear wave equations $$\square {\text{u + u}}\,{\text{ = }}{\text{F(u,u',u''),}}$$ (N.K.G.) , where □ = ∂ t 2 - ∂ 1 2 - ∂ 2 2 -∂ 3 2 is the D’Alembertian of the 4-dimensional Minkowski space-time and F a smooth function of u = u(t,x) and its first and second partial derivates, vanishing, together with its first derivatives, at (u,u′,u″) = 0. In other words, F is at least quadratic in u,u’,u” in a neighborhood of the trivial solution u ≡ 0. We subject u to the pure initial value problem $$ {\text{u = }}{\mkern 1mu} \varepsilon {\text{f}}(x),{{u}_{t}} = \varepsilon g(x)at t = 0 $$ (I.V.P.) with f,g ∈ C 0 ∞ ; (ℝ3) and e > 0, a small parameter.

Uniqueness and nonuniqueness for positive radial solutions of Δu+f(u,r)=0

Abstract: On considere le probleme de l'unicite pour les solutions positives du probleme de Dirichlet non lineaire Δu+f(u,|x|)=0 dans Ω, u/ ∂Ω =0, Ω est une boule ou un anneau de R n et f≥0 est superlineaire en u et satisfaisait f(0,|x|)=0

A gradient bound and a Liouville theorem for nonlinear poisson equations

Abstract: Soit F∈C 2 (R) une fonction non negative et u∈C 3 (R n ) une solution dans tout R n de l'equation Δu=f(u), ou f=F' est la derivee 1ere de F. Si u est borne sur R n et s'il existe x 0 ∈R n tel que F(u(x 0 ))=0, alors u est constante

Brownian motion in a wedge with oblique reflection

Abstract: This work is concerned with the existence and uniqueness of a strong Markov process that has continuous sample paths and the following additional properties: (i) The state space is an infinite two-dimensional wedge, and the process behaves in the interior of the wedge like an ordinary Brownian motion. (ii) The process reflects instantaneously at the boundary of the wedge, the angle of reflection being constant along each side. (iii) The amount of time that the process spends at the comer of the wedge is zero (i.e., the set of times for which the process is at the comer has Lebesgue measure zero). Hereafter, let ξ be the angle of the wedge (0 < ξ < 2π), let θ1 and θ2 be the angles of reflection on the two sides of the wedge, measured from the inward normals, the positive angles being toward the corner (-½π < θ1, θ2 ½π), and set α = (θ1 + θ2)/ξ. The question of existence and uniqueness is recast as a submartingale problem in the style used by Stroock and Varadhan (Diffusion processes with boundary conditions, Comm. Pure Appl. Math. 24, 1971, pp. 147-225), for diffusions on smooth domains with smooth boundary conditions. It is shown that no solution exists if α ≧ 2. In this case, there is a unique continuous strong Markov process satisfying (i)-(ii) above; it reaches the corner of the wedge almost surely and it remains there. If α < 2, however, then there is a unique continuous strong Markov process statisfying (i)-(iii). It is shown that starting away from the corner this process does not reach the corner of the wedge if α ≦ 0, and does reach the corner if 0 < α < 2. The general theory of multi-dimensional diffusions does not apply to the above problem because in general the boundary of the state space is not smooth and there is a discontinuity in the direction of reflection at the corner. For some values of α, the process arises from diffusion approximations to storage systems and queueing networks. (i) The state space is an infinite two-dimensional wedge, and the process behaves in the interior of the wedge like an ordinary Brownian motion. (ii) The process reflects instantaneously at the boundary of the wedge, and the angle of reflection being constant along each side. (iii) The amount of time that the process spends at the corner of the wedge is zero (i.e., the set of times for which the process is at the corner has Lebesgue measure zero).

Inverse scattering and evolution equations

Abstract: On justifie certaines relations formelles bien connues entre les equations d'evolution a 2 variables et les problemes spectraux pour des systemes d'ordre 1 d'equations differentielles. On demontre un resultat d'existence globale pour certaines equations d'evolution

The zero dispersion limit of the korteweg‐de vries equation for initial potentials with non‐trivial reflection coefficient

Abstract: The inverse scattering method is used to determine the distribution limit as ϵ 0 of the solution u(x, t, ϵ) of the initial value problem. Ut − 6uux + ϵ2uxxx = 0, u(x, 0) = v(x), where v(x) is a positive bump which decays sufficiently fast as x x±α. The case v(x) ≪ 0 has been solved by Peter D. Lax and C. David Levermore [8], [9], [10]. The computation of the distribution limit of u(x, t, ϵ) as ϵ 0 is reduced to a quadratic maximization problem, which is then solved.

An L∞ blow‐up estimate for a nonlinear heat equation

Abstract: On decrit une borne superieure au taux d'explosion des solutions de l'equation de la chaleur non lineaire u t =Δu+u p , ou u≥0, p>1

On a weak solution for a transonic flow problem

Abstract: On essaie d'etablir l'existence de solutions faibles pour un ecoulement stationnaire, bidimensionnel, irrotationnel, compressible, non visqueux dans un canal de longueur infinie ou autour d'un profil aerodynamique, la vitesse de l'ecoulement etant donnee a l'infini

Existence of multiple periodic orbits on star-shaped Hamiltonian surfaces

Abstract: Consider the Hamiltonian system (HS) i = 1, …, N. Here, H ϵ C2(ℝ2N, ℝ). In this paper, we investigate the existence of periodic orbits of (HS) on a given energy surface Σ = {z ϵ ℝ2N; H(z) = c} (c > o is a constant). The surface Σ is required to verify certain geometric assumptions: Σ bounds a star-shaped compact region ℛ and αℰ ⊂ ℛ ⊂ βℰ for some ellipsoid ℰ ⊂ ℝ2N, o O (depending in an explicit fashion on the lengths of the main axes of ℰ and one other geometrical parameter of Σ) such that if furthermore β2/α2 < 1 + δ, then (HS) has at least N distinct geometric orbits on Σ. This result is shown to extend and unify several earlier works on this subject (among them works by Weinstein, Rabinowitz, Ekeland-Lasry and Ekeland). In proving this result we construct index theories for an S1 -action, from which we derive abstract critical point theorems for S1 -invariant functionals. We also derive an estimate for the minimal period of solutions to differential equatious.

On the essential spectrum of ideal magnetohydrodynamics. Dedicated to harold grad on the occasion of his sixtieth birthday

Abstract: The essential spectrum of magnetohydrodynamics is shown to arise from waves propagating one-dimensionally along magnetic field lines. Different polarizations of these waves give rise to the Alfven and ballooning spectra. The essential spectrum of an axisymmetric equilibrium when a single azimuthal mode number is considered consists of the Alfven spectrum only, while the ballooning modes appear as intervals of accumulation of discrete eigenvalues with different mode numbers. Some necessary and one sufficient conditions for stability are derived, and some examples where the criteria coincide to yield a simple condition for stability are shown. 29 references.

The generation of modulated wavetrains in the solution of the Korteweg—de vries equation

Abstract: On demontre une relation entre la methode des ondes modulees de Whitham et la methode de Lax et Levermore contenant la limite de petite dispersion

Probabilistic approach to the neumann problem

Abstract: The basic problem considered in this paper is to solve the following Neumann boundary value problem probabilistically: where we assume that q is in a certain functional class to be specified below, and φ is a bounded measurable function on the boundary. We give a martingale formulation of the Neumann problem and show that this formulation is essentially equivalent to the classical formulation. The paper culminates in an explicit formula for the solution of this problem in terms of reflecting Brownian motion and its boundary local time.

Asymptotics of particle trajectories in infinite one‐dimensional systems with collisions

Abstract: We investigate the asymptotic behavior of the trajectories of a tagged particle (tp) in an infinite one-dimensional system of point particles. The particles move independently when not in contact: the only interactions are Harris type generalized elastic collisions which prevent crossings. This is achieved by relabeling the independent trajectories when they cross. When these trajectories are differentiable, as in particles with velocities undergoing Ornstein-Uhlenbeck processes, collisions correspond to exchange of velocities. We prove very generally that the suitably scaled tp trajectory converges (weakly) to a simple Gaussian process. This extends the results of Spitzer for New tonian particles to very general non-crossing processes. The proof is based on the consideration of the simpler process which counts the crossings of the origin by the independent trajectories.

Interfaces with a corner point in one-dimensional porous medium flow

Abstract: We prove that for a large class of initial distributions the solutions of the initial value problem for the one-dimensional porous medium equation have interfaces which start to move abruptly after a positive waiting time. This happens, for example, if the initial pressure is o(|x|2) as x 0. We also give sufficient conditions for the smoothness of the interface which improve previous results.

A theorem of real analysis and its application to free boundary problems

Abstract: In this note we show how comparison theorems for solutions to equations with non-smooth coefficients yield in a very simple fashion some free boundary regularity theorems in cases where there is a priori geometrical information, for instance, when one knows that the solution, u, is monotone in some given direction.

A random-walk interpretation of the incompressible navier-stokes equations

Abstract: We introduce a stochastic process that roughly corresponds to a random walk of a single molecule in an incompressible fluid, and we consider the force (per unit volume) exerted by such a “molecule” on the rest of the fluid. Averaging this force over an ensemble of such molecules and taking an appropriate limit, we obtain the Navier-Stokes equations. The origin of time-irreversibility in these equations is discussed in terms of the model.

The limiting angle of certain Riemannian Brownian motions

Abstract: Let M be a Cartan-Hadamard manifold of dimension d ≧ 3, let p ϵ M and x = exp {r(x)θ(x)} be geodesic polar coordinates with pole p and let X be the Brownian motion on M. Let SectM(x) denote the sectional curvature of any plane section in Mx. We prove that for each c > 2, there is a 0 < β < 1 such that if - L2r(x)2β ≦ SectM(x) ≦ -cr(x)−2 for all x in the complement of a compact set, then limt → ∞ θ(Xt) exists a.s. and defines a nontrivial invariant random variable. The Dirichlet problem at infinity and a conjecture of Greene and Wu are also discussed.

Ohrnstein—uhlenbeck process in a random potential

Abstract: On considere le processus d'Ohmstein-Uhlenbeck sur R d ×R d , dans le potentiel aleatoire -Q~(x,ω). On montre que le processus d'echelle x e (t)=ex(t)e2) obeit au theoreme de la limite centrale en probabilite par rapport a ω