The Bogoljubov-Krylov averaging principle and the Cauchy problem for ordinary differential equations on the half-line

An existence and uniqueness theorem for the Cauchy problem for an ordinary differential equation on the half-line is proved under the hypothesis that the Cauchy problem for the averaged equation has a unique solution. A comparison between the exponential stability of the original equation and the averaged equation is also made. The results established below may be considered as anlogues of the classical Bogoljubov theorem for bounded solutions; they also provide a natural generalization of Mitropol'skij's averaging principle.     

The Fujita exponent for the Cauchy problem in the hyperbolic space

It is well known that the heat kernel in the hyperbolic space has a different behavior for large times than the one in the Euclidean space. The main purpose of this paper is to study its effect on the positive solutions of Cauchy problems with power nonlinearities. Existence and non-existence results for local solutions are derived. Emphasis is put on their long time behavior and on Fujita?s phenomenon. To have the same situation as for the Cauchy problem in RN, namely finite time blow up for all solutions if the exponent is smaller than a critical value and existence of global solutions only for powers above the critical exponent, we must introduce a weight depending exponentially on the time. In this respect the situation is similar to problems in bounded domains with Dirichlet boundary conditions. Important tools are estimates for the heat kernel in the hyperbolic space and comparison principles.     

The Cauchy problem for the elliptic-hyperbolic Davey-Stewartson system in Sobolev space

We study the initial value problem for the elliptic-hyperbolic Davey-Stewartson systems

(0.1)     

The Cauchy problem for the system of thermoelasticity equations in space

We consider the problem of analytic continuation of the solution of the system of thermoelasticity equations in a bounded three-dimensional domain on the basis of known values of the solution and the corresponding stress on a part of the boundary, i.e., the Cauchy problem. We construct an approximate solution of the problem based on the method of Carleman's function.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 212–217, August, 1998.In conclusion, the authors wish to thank Professor M. M. Lavrent'ev and Professor Sh. Ya. Yarmukhamedov for setting the problem and for discussions in the course of the solution.     

The Cauchy problem for a spin-liquid model in three space dimensions

In this paper, we study the Cauchy problem for magnetic fluid of spin-liquid type with Mermin-Ho relation in the three-dimensional space and prove global existence and uniqueness of solutions. The proof is based on the equivalence relation between smooth solutions of the spin-liquid model and the systems of Schrödinger equations with Abelian gauge field. The Sobolev spaces with fractional derivatives are also used in our estimates.     

Fock space on C∞ and Bose-Fock space

In this paper, we introduce the Fock space on ${\mathbb{C}}^{\infty }$ and obtain an isomorphism between the Fock space on ${\mathbb{C}}^{\infty }$ and Bose-Fock space. Based on this isomorphism, we obtain representations of some operators on the Bose-Fock space and answer a question in [2]. As a physical application, we study the Gibbs state.     

Commutative Jacobi fields in Fock space

A jacobi field is understood to be a family (Ã()) of commuting selfadjoint operatorsÃ() acting in a Fock space, having a Jacobi structure, and depending linearly on the test functions . In this article, we give a spectral representation of such a family and outline its applications to the theory of distributions on an infinite dimensional space.This article is dedicated to the memory of my dear teacher Mark G. KreinThe work is partially supported by Fundamental Research Foundation of Ukraine, grant 1.4/62.     

Dixmier trace and the Fock space

We give criteria for products of Toeplitz and Hankel operators on the Fock (Segal–Bargmann) space to belong to the Dixmier class, and compute their Dixmier trace. Along the road, analogous results for the Weyl pseudodifferential operators are also obtained.     

Toeplitz quantization on Fock space

For Toeplitz operators $T_f^{(t)}$ acting on the weighted Fock space $H_t^2$, we consider the semi-commutator $T_f^{(t)}{T}_g^{(t)}?T_{fg}^{(t)}$, where $t>0$ is a certain weight parameter that may be interpreted as Planck's constant ? in Rieffel's deformation quantization. In particular, we are interested in the semi-classical limit

($?$)$\underset{t\to 0}{\lim }?{6T_f^{(t)}{T}_g^{(t)}?T_{fg}^{(t)}6}_t.$

It is well-known that ${6T_f^{(t)}{T}_g^{(t)}?T_{fg}^{(t)}6}_t$ tends to 0 under certain smoothness assumptions imposed on f and g. This result was recently extended to $f,g\in \mathrm{BUC}({\mathbb{C}}^n)$ by Bauer and Coburn. We now further generalize (?) to (not necessarily bounded) uniformly continuous functions and symbols in the algebra $\mathrm{VMO}\cap L^{\infty }$ of bounded functions having vanishing mean oscillation on ${\mathbb{C}}^n$. Our approach is based on the algebraic identity $T_f^{(t)}{T}_g^{(t)}?T_{fg}^{(t)}=?{(H_{\overline{f}}^{(t)})}^{?}{H}_g^{(t)}$, where $H_g^{(t)}$ denotes the Hankel operator corresponding to the symbol g, and norm estimates in terms of the (weighted) heat transform. As a consequence, only f (or likewise only g) has to be contained in one of the above classes for (?) to vanish. For g we only have to impose ${\lim \sup }_{t\to 0}{6H_g^{(t)}6}_t<\infty$, e.g. $g\in L^{\infty }({\mathbb{C}}^n)$. We prove that the set of all symbols $f\in L^{\infty }({\mathbb{C}}^n)$ with the property that ${\lim }_{t\to 0}?{6T_f^{(t)}{T}_g^{(t)}?T_{fg}^{(t)}6}_t={\lim }_{t\to 0}?{6T_g^{(t)}{T}_f^{(t)}?T_{gf}^{(t)}6}_t=0$ for all $g\in L^{\infty }({\mathbb{C}}^n)$ coincides with $\mathrm{VMO}\cap L^{\infty }$. Additionally, we show that $\underset{t\to 0}{\lim }?{6T_f^{(t)}6}_t={6f6}_{\infty }$ holds for all $f\in L^{\infty }({\mathbb{C}}^n)$. Finally, we present new examples, including bounded smooth functions, where (?) does not vanish.     

On a complete discretization scheme for an ill-posed Cauchy problem in a Banach space

A complete discretization scheme for an ill-posed Cauchy problem for abstract firstorder linear differential equations with sectorial operators in a Banach space is validated. The scheme combines a time semidiscretization of the equations and a finite-dimensional approximation of the spaces and operators. Regularization properties of the scheme are established. Error estimates are obtained in the case of approximate initial data under various a priori assumptions concerning the solution.