Support theorem for the fundamental solution to the Schrödinger equation on certain compact symmetric spaces

In this paper we construct the fundamental solution to the Schödinger equation on a compact symmetric space with even root multiplicities using shift operators of Heckman and Opdam. Next, we prove that the support of the fundamental solution becomes a lower dimensional subset at a rational time whereas its support and its singular support coincide with the whole symmetric space at an irrational time. Moreover, we also show that generalized Gauss sums appear in the expression of the fundamental solution.     

Asymptotic behavior for the filtration equation in domains with noncompact boundary

We consider the initial value boundary problem with zero Neumann data for an equation modeled after the porous media equation, with variable coefficients. The spatial domain is unbounded and shaped like a (general) paraboloid, and the solution u is integrable in space and nonnegative. We show that the asymptotic profile for large times of u is one dimensional and given by an explicit function, which can be regarded as the fundamental solution of a one-dimensional differential equation with weights. In the case when the domain is a cone or the whole space (Cauchy problem), we obtain a genuine multidimensional profile given by the well-known Barenblatt solution.     

$ L^p $-Theory of the Stokes equation in a half space

In this paper, we investigate -estimates for the solution of the Stokes equation in a half space H where . It is shown that the solution of the Stokes equation is governed by an analytic semigroup on or . From the operatortheoretical point of view it is a surprising fact that the corresponding result for does not hold true. In fact, there exists an -function f satisfying such that the solution of the corresponding resolvent equation with right hand side f does not belong to . Taking into account however a recent result of Kozono on the nonlinear Navier-Stokes equation, the -result is not surprising and even natural. We also show that the Stokes operator admits a R-bounded -calculus on for 1 < p < and obtain as a consequence maximal -regularity for the solution of the Stokes equation. Received August 24, 2000; accepted September 30, 2000.     

Necessary and sufficient condition for the stabilization of the solution of the cauchy problem for a special heat equation

In the space ℝ ⁿ , we obtain the solution of the Cauchy problem for the equation degenerating at the origin, where and u _rr ″ is the second derivative in the direction of the position vector of the point . We study the stabilization of this solution.     

A Periodic Problem for the Landau--Ginzburg Equation

In this paper, we consider a periodic problem for the n-dimensional complex Landau--Ginzburg equation. It is shown that in the case of small initial data there exists a unique classical solution of this problem, and an asymptotics of this solution uniform in the space variable is given. The leading term of the asymptotics is exponentially decreasing in time.     

Unique solvability of the inverse problem of determination of the leading coefficient in a parabolic equation

We study existence and uniqueness of the solution for the inverse problem of determination of the unknown coefficient ϱ(t) multiplying u _t in a nondivergence parabolic equation. As additional information, the integral of the solution over the domain of space variables with some given weight function is specified. The coefficients of the equation depend both on time and on the space variables.     

On the Neumann problem for the Helmholtz equation in a plane angle

We prove that the solution of the Neumann problem for the Helmholtz equation in a plane angle Ω with boundary conditions from the space H_−1/2(Γ), where Γ is the boundary of Ω, which is provided by the well‐known Sommerfeld integral, belongs to the Sobolev space H₁(Ω) and depends continuously on the boundary values. To this end, we use another representation of the solution given by the inverse two‐dimensional Fourier transform of an analytic function depending on the Cauchy data of the solution. Copyright © 2000 John Wiley & Sons, Ltd.     

Mixing “In the sense of Ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. I

We prove that a bounded 1-periodic function of a solution of a time-homogeneous diffusion equation with 1-periodic coefficients forms a process that satisfies the condition of uniform strong mixing. We obtain an estimate for the rate of approach of a certain normalized integral functional of a solution of an ordinary time-homogeneous stochastic differential equation with 1-periodic coefficients to a family of Wiener processes in probability in the metric of space C [0, T]. As an example, we consider an ordinary differential equation perturbed by a rapidly oscillating centered process that is a 1-periodic function of a solution of a time-homogeneous stochastic differential equation with 1-periodic coefficients. We obtain an estimate for the rate of approach of a solution of this equation to a solution of the corresponding It? stochastic equation.     

Existence and Smoothness of the Density for Spatially Homogeneous SPDEs

In this paper, we extend Walsh’s stochastic integral with respect to a Gaussian noise, white in time and with some homogeneous spatial correlation, in order to be able to integrate some random measure-valued processes. This extension turns out to be equivalent to Dalang’s one. Then we study existence and regularity of the density of the probability law for the real-valued mild solution to a general second order stochastic partial differential equation driven by such a noise. For this, we apply the techniques of the Malliavin calculus. Our results apply to the case of the stochastic heat equation in any space dimension and the stochastic wave equation in space dimension d=1,2,3. Moreover, for these particular examples, known results in the literature have been improved.      

Singular Perturbation of Kolmogorov Equation in Gauss–Sobolev Space

Abstract

This article is concerned with the Kolmogorov equation associated to a stochastic partial differential equation with an additive noise depending on a small parameter ε > 0. As ε vanishes, the parabolic equation degenerates into a first-order evolution equation. In a Gauss–Sobolev space setting, we prove that, as ε ↓ 0, the solution of the Cauchy problem for the Kolmogorov equation converges in L ²(μ, H) to that of the reduced evolution equation of first-order, where μ is a reference Gaussian measure on the Hilbert space H.     

Dirac-Lu space with pseudo-Riemannian metric of constant curvature

In this paper, we will discuss the geometries of the Dirac-Lu space whose boundary is the third conformal space N defined by Dirac and Lu. We firstly give the SO(3, 3) invariant pseudo-Riemannian metric with constant curvature on this space, and then discuss the timelike geodesics. Finally we get a solution of the Yang-Mills equation on it by using the reduction theorem of connections.     

On the Cauchy problem for Wigner–Poisson–BGK equation in the Wiener algebra

In this paper, we prove the existence and uniqueness of the local mild solution to the Cauchy problem of the n‐dimensional (n≥3) Wigner–Poisson–BGK equation in the space of some integrable functions whose inverse Fourier transform are integrable. The main difficulties in establishing mild solution are to derive the boundedness and locally Lipschitz properties of the appropriate nonlinear terms in the Wiener algebra. Copyright © 2015 John Wiley & Sons, Ltd.     

Finite-dimensional approximations for the equation of nonlinear filtering derived in mild form

The Zakai equation for the unnormalized conditional density is derived as a mild stochastic bilinear differential equation on a suitableL ² space. It is assumed that the Markov semigroup corresponding to the state process isC ₀ on such space. This allows the establishment of the existence and uniqueness of the solution by means of general theorems on stochastic differential equations in Hilbert space. Moreover, an easy treatment of convergence conditions can be given for a general class of finite-dimensional approximations, including Galerkin schemes. This is done by using a general continuity result for the solution of a mild stochastic bilinear differential equation on a Hilbert space with respect to the semigroup, the forcing operator, and the initial state, within a suitable topology.     

Logarithmic derivatives of invariant measure for stochastic differential equations in hilbert spaces

We consider a process X solution of a semilinear stochastic evolution equation in a Hilbert space. Assuming that X has an invariant measure ν, we investigate its regularity properties. Logarithmic derivatives of ν in certain directions, are shown to exist under appropriate conditions on the nonlinear term in the equation. A set of directions of differentiability for ν is explicitly described in terms of the coefficients of the equation. In some cases, logarithmic derivatives are represented as conditional expectations of random variables related to an appropriate stationary process. An application to a system of stochastic partial differential equations in one space variable is given     

A convergence analysis and applications for the Newton-Kantorovich method in K-normed spaces

We study the local and semilocal convergence of the Newton-Kantorovich method to a solution of a nonlinear operator equation on aK-normed space setting. Using more precise majorizing sequences than before we show that in the semilocal case finer error bounds can be determined on the distances involved and an at least as precise information on the location of the solution as in earlier results. In the local case we show that a larger radius of convergence can be obtained.     

New bounds for the Kuramoto‐Sivashinsky equation

We show that every L‐periodic mean‐zero solution u of the Kuramoto‐Sivashinsky equation is on average o(L) for L ? 1, in the sense that for any T > 0 the space average of | u(t) | is bounded by for any t > T and any L sufficiently large. For this we argue that on large spatial scales, the solution behaves like an entropy solution of the inviscid Burgers equation. The analysis of this non‐standard perturbation of the Burgers equation is based on a “div‐curl” argument. © 2004 Wiley Periodicals, Inc.     

Asymptotics of the density matrix for a system of a large number of identical particles

The Wigner equation is considered for a system of a large numberN of identical particles with interaction factor of the order of 1/N. In both the Bose and the Fermi cases, we construct the asymptotics of the solution of the Cauchy problem for this equation with regard to the exchange effect for the case in which the Planck constant is of the order ofN ^−1/d , whered is the space dimension. This asymptotics is interpreted in terms of the theory of the complex germ on a curved phase space equivalent to the space of functions with values on the Riemann sphere in the Fermi case and on the Lobachevskii plane in the Bose case. The classical equations of motion in both cases are reduced to the Vlasov equation; since the phase space is infinite-dimensional, the complex germ is subjected to additional conditions depending on the type of statistics. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 84–106, January, 1999.     

Linear differential equations in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> with radial derivatives

We obtain a solution of a linear differential equation with a radial derivative. The coefficients and the solution are functionals on L ₂[a, b]. In the same space, we study the properties of solutions of second-order linear homogeneous equations.     

A Mixed Problem with Integral Condition for the Hyperbolic Equation

In this paper, we study a mixed problem for the hyperbolic equation with a boundary Neumann condition and a nonlocal integral condition. We justify the assertion that there exists a unique generalized solution of the problem under consideration. The proof of uniqueness is based on an estimate, derived a priori, in the function space introduced in the paper, while the existence of a generalized solution is proved by the Galerkin method.     

On the validity of the Ginzburg-Landau equation

Summary The famous Ginzburg-Landau equation describes nonlinear amplitude modulations of a wave perturbation of a basic pattern when a control parameterR lies in the unstable regionO(ε ²) away from the critical valueR _c for which the system loses stability. Hereε>0 is a small parameter. G-L's equation is found for a general class of nonlinear evolution problems including several classical problems from hydrodynamics and other fields of physics and chemistry. Up to now, the rigorous derivation of G-L's equation for general situations is not yet completed. This was only demonstrated for special types of solutions (steady, time periodic) or for special problems (the Swift-Hohenberg equation). Here a mathematically rigorous proof of the validity of G-L's equation is given for a general situation of one space variable and a quadratic nonlinearity. Validity is meant in the following sense. For each given initial condition in a suitable Banach space there exists a unique bounded solution of the initial value problem for G-L's equation on a finite interval of theO(1/ε²)-long time scale intrinsic to the modulation. For such a finite time interval of the intrinsic modulation time scale on which the initial value problem for G-L's equation has a bounded solution, the initial value problem for the original evolution equation with corresponding initial conditions, has a unique solutionO(ε²) — close to the approximation induced by the solution of G-L's equation. This property guarantees that, for rather general initial conditions on the intrinsic modulation time scale, the behavior of solutions of G-L's equation is really inherited from solutions of the original problem, and the other way around: to a solution of G-L's equation corresponds a nearby exact solution with a relatively small error.