Construction of strong solutions of SDE's via Malliavin calculus

In this paper we develop a new method for the construction of strong solutions of stochastic equations with discontinuous coefficients. We illustrate this approach by studying stochastic differential equations driven by the Wiener process. Using Malliavin calculus we derive the result of A.K. Zvonkin (1974) [31] for bounded and measurable drift coefficients as a special case of our analysis of SDE's. Moreover, our approach yields the important insight that the solutions obtained by Zvonkin are even Malliavin differentiable. The latter indicates that the “nature” of strong solutions of SDE's is tightly linked to the property of Malliavin differentiability. We also stress that our method does not involve a pathwise uniqueness argument but provides a direct construction of strong solutions.     

Strong p-completeness of stochastic differential equations and the existence of smooth flows on noncompact manifolds

Summary Here we discuss the regularity of solutions of SDE's and obtain conditions under which a SDE on a complete Riemannian manifoldM has a global smooth solution flow, in particular improving the usual global Lipschitz hypothesis whenM=R ⁿ . There are also results on non-explosion of diffusions.Research supported by SERC grant GR/H67263     

On the existence of solutions with smooth density of stochastic differential equations in plane

In this paper we apply the Malliavin calculus for two-parameter Wiener functionals to show that the solutions of stochastic differential equations in plane have a smooth density under the restricted Hörmander's condition. This answers a question mentioned by Nualart and Sanz in [3].     

随机微分方程数值解法

§1.前言 设?_t为(Ω,?,P)上的m维布朗运动(简记为BM).?_t≡σ(B_s;s≤t),于是可在(Ω,?_t,?,P)上定义随机微分方程(记成SDE) ?其中?∈R~n,?是n×m矩阵. 方程(1.1)在物理、化学、生物学等各种不同领域有着重要的应用;就数学本身而言,它在微分方程、控制论、非线性滤波中的作用也日益显著.因此,SDE的数值解法的研究,引起人们的广泛注意.本文研究的是?=1的数值解法,对一般情形,也可完全类似地得到一系列结果,只是数值解具有不同的精度.本文仅给出一维结果,多维情形平行可得.     

On the Regularity of Distribution for a Solution of SDE of a Jump Type with Arbitrary Levy Measure of the Noise

The local properties of distributions of solutions of SDE's with jumps are studied. Using the method based on the “time-wise” differentiation on the space of functionals of Poisson point measure, we give a full analog of Hormander condition, sufficient for the solution to have a regular distribution. This condition is formulated only in terms of coefficients of the equation and does not require any regularity properties of the Levy measure of the noise. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 9, pp. 1261–1283, September, 2005.     

Approximation by solutions of elliptic equations in semilocal spaces

In the present work, we investigate the approximability of solutions of elliptic partial differential equations in a bounded domain Ω by solutions of the same equations in a larger domain. We construct an abstract framework which allows us to deal with such density questions, simultaneously for various norms. More specifically, we study approximations with respect to the norms of semilocal Banach spaces of distributions. These spaces are required to satisfy certain postulates. We establish density results for elliptic operators with constant coefficients which unify and extend previous results. In our density results Ω may possess holes and it is required to satisfy the segment condition. We observe that analogous density results do not hold in spaces where the infinitely smooth functions are not dense. Finally, we provide applications related to the method of fundamental solutions.     

On the Malliavin Calculus for SDE's on Hilbert Spaces

We consider a class of SDE's on Hilbert spaces and study the partial hypoellipticity of generators associated with these SDE's. We show that the Malliavin calculus with a new key lemma is efficient for the purpose. The partial Hörmander theorem is proved in this paper, and it is applied to the problem of propagation of absolute continuity of measures by stochastic flows given by those SDE's.     

Blow-up of smooth solutions to the compressible fluid models of Korteweg type

We show the blow-up of smooth solutions to a non-isothermal model of capillary compressible fluids in arbitrary space dimensions with initial density of compact support. This is an extension of Xin’s result [Xin, Z.: Blow-up of smooth solutions to the compressible Navier-Stokes equations with compact density. Comm. Pure Appl. Math., 51, 229–240 (1998)] to the capillary case but we do not need the condition that the entropy is bounded below. Moreover, from the proof of Theorem 1.2, we also obtain the exact relationship between the size of support of the initial density and the life span of the solutions. We also present a sufficient condition on the blow-up of smooth solutions to the compressible fluid models of Korteweg type when the initial density is positive but has a decay at infinity.     

On solutions of stochastic differential equations with drift

Summary We study stochastic differential equations of the formdX _t=(X _t)dM_t+b(X_t)d_t whereM is a continuous local martingale and <M> stands for its quadratic variation process. The conditions introduced by Engelbert and Schmidt, which ensure the existence and uniqueness in law of solutions of SDE's driven by the Wiener process without drift (or with generalized drift) are shown to be no longer valid.     

Global gradient estimates for degenerate parabolic equations in nonsmooth domains

This paper studies the global regularity theory for degenerate nonlinear parabolic partial differential equations. Our objective is to show that weak solutions belong to a higher Sobolev space than assumed a priori if the complement of the domain satisfies a capacity density condition and if the boundary values are sufficiently smooth. Moreover, we derive integrability estimates for the gradient. The results extend to the parabolic systems as well. The higher integrability estimates provide a useful tool in several applications.      

Hyperfunctions and (analytic) hypoellipticity

In this work we discuss the problem of smooth and analytic regularity for hyperfunction solutions to linear partial differential equations with analytic coefficients. In particular we show that some well known “sum of squares” operators, which satisfy Hörmander’s condition and consequently are hypoelliptic, admit hyperfunction solutions that are not smooth (in particular they are not distributions).     

Nonlinear behavior of model equations which are linearly ill-posed

We discuss two model equations of nonlinear evolution which demonstrate that linearly ill-posed problems may be well-posed in a mild sense. For the nonlocal equation (1.4), smooth solutions exist for all time, are unique, and depend continuously on the initial data in low norms. For the partial differential equation (1.1), solutions always exist; we do not know whether they are unique, but if they are, they also have continuous dependence on data. The large-time behavior of solutions and other qualitative properties are discussed     

Stochastic functional differential equations on manifolds

In this paper, we study stochastic functional differential equations (sfde's) whose solutions are constrained to live on a smooth compact Riemannian manifold. We prove the existence and uniqueness of solutions to such sfde's. We consider examples of geometrical sfde's and establish the smooth dependence of the solution on finite-dimensional parameters. Received: 6 July 1999 / Revised version: 19 April 2000 /?Published online: 14 June 2001     

一类流固耦合系统强解的存在性问题

本文主要讨论一类多刚体与粘性系数依赖于密度的不可压缩流体耦合系统的强解存在性问题.首先,利用变量替换建立本文研究对象对应的非线性微分方程,然后,利用Garlerkin逼近方法获得线性化问题的光滑解,从而可以构造出原问题的逼近解.通过估计逼近解的一致有界性,最后证明了一类描述多刚体在不可压缩流体中运动的耦合系统强解的存在性问题.特别要指出的是在初始密度满足相容性条件下,此结论允许流体内部出现真空.     

Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density

We present a sufficient condition on the blowup of smooth solutions to the compressible Navier-Stokes equations in arbitrary space dimensions with initial density of compact support. As an immediate application, it is shown that any smooth solutions to the compressible Navier-Stokes equations for polytropic fluids in the absence of heat conduction will blow up in finite time as long as the initial densities have compact support, and an upper bound, which depends only on the initial data, on the blowup time follows from our elementary analysis immediately. Another implication is that there is no global small (decay in time) or even bounded (in the case that all the viscosity coefficients are positive) smooth solutions to the compressible Navier-Stokes equations for polytropic fluids, no matter how small the initial data are, as long as the initial density is of compact support. This is in contrast to the classical theory of global existence of small solutions to the same system with initial data being a small perturbation of a constant state that is not a vacuum. The blowup of smooth solutions to the compressible Euler system with initial density and velocity of compact support is a simple consequence of our argument. © 1998 John Wiley & Sons, Inc.     

Properties of positive solutions to an elliptic equation with negative exponent

In this paper, we study some quantitative properties of positive solutions to a singular elliptic equation with negative power on the bounded smooth domain or in the whole Euclidean space. Our model arises in the study of the steady states of thin films and other applied physics as well as differential geometry. We can get some useful local gradient estimate and L¹ lower bound for positive solutions of the elliptic equation. A uniform positive lower bound for convex positive solutions is also obtained. We show that in lower dimensions, there is no stable positive solutions in the whole space. In the whole space of dimension two, we can show that there is no positive smooth solution with finite Morse index. Symmetry properties of related integral equations are also given.     

Formation of singularities in solutions to the compressible radiation hydrodynamics equations with vacuum

We study the Cauchy problem for multi-dimensional compressible radiation hydrodynamics equations with vacuum. First, we present some sufficient conditions on the blow-up of smooth solutions in multi-dimensional space. Then, we obtain the invariance of the support of density for the smooth solutions with compactly supported initial mass density by the property of the system under the vacuum state. Based on the above-mentioned results, we prove that we cannot get a global classical solution, no matter how small the initial data are, as long as the initial mass density is of compact support. Finally, we will see that some of the results that we obtained are still valid for the isentropic flows with degenerate viscosity coefficients as well as for one-dimensional case.     

Spaces of Quasi-Periodic Functions and Oscillations in Differential Equations

We build spaces of q.p. (quasi-periodic) functions and we establish some of their properties. They are motivated by the Percival approach to q.p. solutions of Hamiltonian systems. The periodic solutions of an adequatez partial differential equation are related to the q.p. solutions of an ordinary differential equation. We use this approach to obtain some regularization theorems of weak q.p. solutions of differential equations. For a large class of differential equations, the first theorem gives a result of density: a particular form of perturbated equations have strong solutions. The second theorem gives a condition which ensures that any essentially bounded weak solution is a strong one.     

Smooth solutions of non-linear stochastic partial differential equations driven by multiplicative noises

In this paper, we study the regularity of solutions of nonlinear stochastic partial differential equations (SPDEs) with multiplicative noises in the framework of Hilbert scales. Then we apply our abstract result to several typical nonlinear SPDEs such as stochastic Burgers and Ginzburg-Landau equations on the real line, stochastic 2D Navier-Stokes equations (SNSEs) in the whole space and a stochastic tamed 3D Navier-Stokes equation in the whole space, and obtain the existence of their smooth solutions respectively. In particular, we also get the existence of local smooth solutions for 3D SNSEs.     

Implicit Second Order Differential Inclusions in Reflexive Smooth Banach Spaces

In this paper we prove the existence of solutions of some types of second order implicit differential inclusions in reflexive smooth Banach spaces.