An approximate method via Taylor series for stochastic functional differential equations

The subject of this paper is an analytic approximate method for stochastic functional differential equations whose coefficients are functionals, sufficiently smooth in the sense of Fréchet derivatives. The approximate equations are defined on equidistant partitions of the time interval, and their coefficients are general Taylor expansions of the coefficients of the initial equation. It will be shown that the approximate solutions converge in the Lp-norm and with probability one to the solution of the initial equation, and also that the rate of convergence increases when degrees in Taylor expansions increase, analogously to real analysis.     

An analytic approximation of solutions of stochastic differential delay equations with Markovian switching

In this paper, we are concerned with the stochastic differential delay equations with Markovian switching (SDDEwMSs). As stochastic differential equations with Markovian switching (SDEwMSs), most SDDEwMSs cannot be solved explicitly. Therefore, numerical solutions, such as EM method, stochastic Theta method, Split-Step Backward Euler method and Caratheodory’s approximations, have become an important issue in the study of SDDEwMSs. The key contribution of this paper is to investigate the strong convergence between the true solutions and the numerical solutions to SDDEwMSs in the sense of the L^p-norm when the drift and diffusion coefficients are Taylor approximations.     

An analytic approximate method for solving stochastic integrodifferential equations

In this paper we compare the solution of a general stochastic integrodifferential equation of the Ito type, with the solutions of a sequence of appropriate equations of the same type, whose coefficients are Taylor series of the coefficients of the original equation. The approximate solutions are defined on a partition of the time-interval. The rate of the closeness between the original and approximate solutions is measured in the sense of the Lp-norm, so that it decreases if the degrees of these Taylor series increase, analogously to real analysis. The convergence with probability one is also proved.     

Pointwise estimates for the fundamental solutions of a class of singular parabolic problems

We deal with the Cauchy problem associated to a class of quasilinear singular parabolic equations with L ^∞ coefficients whose prototypes are the p-Laplacian (2N/(N + 1) < p < 2) and the porous medium equation (((N ? 2)/N)₊ < m < 1). We prove existence of and sharp pointwise estimates from above and from below for the fundamental solutions. Our results can be extended to general non-negative L ¹ initial data.     

Integrated semigroups and linear partial differential equations with delay

We study existence and uniqueness of solutions for linear partial differential equations with delay in L^p-spaces using an approach of Batkai and Piazzera and a recent perturbation result for integrated semigroups. We apply our result to an equation with delay in the highest-order derivatives.     

Estimates of best approximations in integral metrics and Fourier coefficients with respect to multiplicative systems

We obtain estimates of approximations by angle in Hardy and L ^p spaces in terms of double Fourier-Vilenkin coefficients. Analogous results are also established for best approximations in the one-dimensional case.     

Orthogonal wavelets on direct products of cyclic groups

We describe a method for constructing compactly supported orthogonal wavelets on a locally compact Abelian group G which is the weak direct product of a countable set of cyclic groups of pth order. For all integers p, n ≥ 2, we establish necessary and sufficient conditions under which the solutions of the corresponding scaling equations with p ⁿ numerical coefficients generate multiresolution analyses in L ²(G). It is noted that the coefficients of these scaling equations can be calculated from the given values of p ⁿ parameters using the discrete Vilenkin-Chrestenson transform. Besides, we obtain conditions under which a compactly supported solution of the scaling equation in L ²(G) is stable and has a linearly independent system of “integer” shifts. We present several examples illustrating these results.     

ONE LINEAR ANALYTIC APPROXIMATION FOR STOCHASTIC INTEGRODIFFERENTIAL EQUATIONS

This article concerns the construction of approximate solutions for a general stochastic integrodifferential equation which is not explicitly solvable and whose coeffcients functionally depend on Lebesgue integrals and stochastic integrals with respect to martingales. The approximate equations are linear ordinary stochastic differential equations, the solutions of which are defined on sub-intervals of an arbitrary partition of the time interval and connected at successive division points. The closeness of the initial and approximate solutions is measured in the L^p-th norm, uniformly on the time interval. The convergence with probability one is also given.     

Parabolic equations in time dependent Reifenberg domains

In this paper we define time dependent parabolic Reifenberg domains and study Lp estimates for weak solutions of uniformly parabolic equations in divergence form on these domains. The basic assumption is that the principal coefficients are of parabolic BMO space with small parabolic BMO seminorms. It is shown that Lp estimates hold for time dependent parabolic δ-Reifenberg domains.     

The Cauchy problem for critical and subcritical semilinear parabolic equations in Lr (II), initial data in critical Sobolev spaces Ḣ−s,rs

By presenting some time–space L^p−L^r estimates, we will establish the local and global existence and uniqueness of solutions for semilinear parabolic equations with the Cauchy data in critical Sobolev spaces of negative indices. Our results contain the complex (derivative) Ginzburg–Landau equation and the Cahn–Hilliard equation as special cases.     

Extinction in nonautonomous competitive Lotka-Volterra systems with infinite delay

The principle of competitive exclusion is extended to n-species nonautonomous Lotka-Volterra competition systems of differential equations with infinite delay. It is shown that if the coefficients are bounded, continuous and satisfy certain inequalities, then any solution with initial function in an appropriate space will have n−1 of its components tend to zero, while the remaining one will stabilize at a certain solution of a logistic differential equation.     

Regularization for a class of ill-posed evolution problems in Banach space

We study evolution equations in Banach space, and provide a general framework for regularizing a wide class of ill-posed Cauchy problems by proving continuous dependence on modeling for nonautonomous equations. We approximate the ill-posed problem by a well-posed one, and obtain H?lder-continuous dependence results that provide estimates of the error for a class of solutions under certain stabilizing conditions. For examples that include the linearized Korteweg-de Vries equation and the Schr?dinger equation in L ^p ,p??2, we obtain a family of regularizing operators for the ill-posed problem. This work extends to the nonautonomous case several recent results for ill-posed problems with constant coefficients.     

FBSDEs with time delayed generators: L-solutions, differentiability, representation formulas and path regularity

We extend the work of Delong and Imkeller (2010) [6] and [7] concerning backward stochastic differential equations with time delayed generators (delay BSDEs). We give moment and a priori estimates in general L^p-spaces and provide sufficient conditions for the solution of a delay BSDE to exist in L^p. We introduce decoupled systems of SDEs and delay BSDEs (delay FBSDEs) and give sufficient conditions for their variational differentiability. We connect these variational derivatives to the Malliavin derivatives of delay FBSDEs via the usual representation formulas. We conclude with several path regularity results, in particular we extend the classic L²-path regularity to delay FBSDEs.     

Coefficient estimates for negative powers of the derivative of univalent functions

Letf be a one-to-one analytic function in the unit disc withf′(0)=1. We prove sharp estimates for certain Taylor coefficients of the functions(f′) ^p , wherep<0. The proof is similar to de Branges’ proof of Bieberbach’s conjecture, and thus relies on Löwner’s equation. A special case leads to a generalization of the usual estimate for the Schwarzian derivative off. We use this to improve known estimates for integral means of the functions |f′| ^p for integersp??2.     

Local convergence of the method of pseudolinear equations for quasilinear elliptic boundary-value problems

A variant of the method of pseudolinear equations, an iterative method of solving quasilinear partial differential equations, is described for quasilinear elliptic boundary-value problems of the type -[p₁(u_x)]_x - [p₂(u_y)]_y = f on a bounded simply connected two-dimensional domain D. A theorem on local convergence in C^{2, λ}(D) of this variant, which has constant coefficients, is proved. Three other method of solving quasilinear elliptic boundary-value problems, namely. Newton's method, the Ka?anov method and a variant of the method of successive approximations that has constant coefficients, are briefly discussed. Results of a series of numerical experiments in a finite-difference setting of solving quasilinear Dirichlet problems of the above-mentioned type by the method of pseudolinear equations and these three methods are given. These results show that Newton's method converges for stronger nonlinearities than do the other methods, which, in order thereafter, are the Ka?anov method, the method of pseudolinear equations and, last, the method of successive approximations, which converges only for relatively weak nonlinearities. From fastest to slowest, the methods are: the method of successive approximations, the method of pseudolinear equations, Newton's method, the Ka?anov method.     

A Taylor polynomial approach for solving differential-difference equations

The purpose of this study is to give a Taylor polynomial approximation for the solution of mth-order linear differential-difference equations with variable coefficients under the mixed conditions about any point. For this purpose, Taylor matrix method is introduced. This method is based on first taking the truncated Taylor expansions of the functions in the differential-difference equations and then substituting their matrix forms into the equation. Hence, the result matrix equation can be solved and the unknown Taylor coefficients can be found approximately. In addition, examples that illustrate the pertinent features of the method are presented, and the results of study are discussed. Also we have discussed the accuracy of the method. We use the symbolic algebra program, Maple, to prove our results.     

The homogeneous dirichlet problem for quasilinear anisotropic degenerate parabolic-hyperbolic equation with L initial value

The aim of this paper is to prove the well-posedness (existence and uniqueness) of the L^p entropy solution to the homogeneous Dirichlet problems for the anisotropic degenerate parabolic-hyperbolic equations with L^p initial value. We use the device of doubling variables and some technical analysis to prove the uniqueness result. Moreover we can prove that the L^p entropy solution can be obtained as the limit of solutions of the corresponding regularized equations of nondegenerate parabolic type.     

Existence and stability of planar diffusion waves for 2-D Euler equations with damping

To study the non-linear stability of a non-trivial profile for a multi-dimensional systems of gas dynamics, the combination of the Green function on estimating the lower order derivatives and the energy method for the higher order derivatives is shown to be not only useful but sometimes maybe also essential. In this paper, we study the stability of a planar diffusion wave for the isentropic Euler equations with damping in two-dimensional space. By introducing an approximate Green function for the linearized equations around the planar diffusion wave and by applying the energy method, we prove the global existence and the L₂ convergence rate of the solution when the initial data is a small perturbation of the planar diffusion wave. The decay rates of the perturbation and its lower order spatial derivatives obtained are optimal in the L₂ norm. Furthermore, the constructed approximate Green function in this paper can be used for the pointwise and the Lp estimates of the solutions concerned. In fact, the approach by combining of the Green function and energy method can be applied to other system especially when the derivatives of the coefficients in the system have certain time decay properties.     

A probabilistic interpretation and stochastic particle approximations of the 3-dimensional Navier-Stokes equations

We develop a probabilistic interpretation of local mild solutions of the three dimensional Navier-Stokes equation in the L^p spaces, when the initial vorticity field is integrable. This is done by associating a generalized nonlinear diffusion of the McKean-Vlasov type with the solution of the corresponding vortex equation. We then construct trajectorial (chaotic) stochastic particle approximations of this nonlinear process. These results provide the first complete proof of convergence of a stochastic vortex method for the Navier-Stokes equation in three dimensions, and rectify the algorithm conjectured by Esposito and Pulvirenti in 1989. Our techniques rely on a fine regularity study of the vortex equation in the supercritical L^p spaces, and on an extension of the classic McKean-Vlasov model, which incorporates the derivative of the stochastic flow of the nonlinear process to explain the vortex stretching phenomenon proper to dimension three. Supported by Fondecyt Project 1040689 and Nucleus Millennium Information and Randomness ICM P01-005.