An Essay On Well-posedness Of Stochastic(Partial)Differential Equations With Singular Coefficients And Related Problems

Stochastic differential equations have become a hot topic research field since It ?o's time.Many mathematicians such as Kolmogorov?It ?o?Krylov are interested in stochastic differential equations.In the disciplinary fields,such as population growth system,filter system,prey-predator system,these systems can be modeled by stochastic differential equations so that many scientists in these research subjects are also interested in stochastic differential equations,new research results appear successively in this area.This dissertation starts from stochastic differential equations,which mainly considers the well-posedness of(non-local)stochastic partial differential equations related to stochastic differential equations and the related problems.The layout of this thesis is as follows.We introduce in Chapter 1 the physical background and research progress on this thesis,and review some preliminary knowledge,including common symbols and some classical inequalities,function spaces and so on.In Chapter 2 we deal with the Lipschitz and W2,?estimates for a second-order parabolic PDEs with zero initial data and nonlinear term satisfing a Ladyzhenskaya-Prodi-Serrin type condition.Moreover,we give two applications of the theoretic results.We first discuss the regularity of the stochastic heat equations,and then we consider the Sobolev differentiability of strong solutions to a class of stochastic differential equations with singular drift coefficients.In Chapter 3 we study the existence and uniqueness of stochastic entropy solution for a nonlinear transport equation with a stochastic perturbation.The uniqueness is based on the doubling variable method.For the existence we develop a new scheme of parabolic approximation motivated by the method of vanishing viscosity.Furthermore,we prove the continuous dependence of stochastic strong entropy solutions on the coefficient b and the nonlinear function f.In Chapter 4 we consider a class of nonlocal elliptic equations on a bounded domain.By the Lax-Milgram theorem and the De Giorgi iteration technique,we prove the existenceand uniqueness of L?solution.Furthermore,we investigate the existence of densities for measure-valued solutions to nonhomogeneous measure-valued nonlocal elliptic equations.In Chapter 5 we study the nonlocal partial differential equations driven by additive white noises.For Lqintegrable coefficients,we establish the existence and uniqueness,as well as H ¨older continuous of mild solutions.With the help of tail estimation method,we get the H ¨older estimations.In the end,we give some problems for further research on the related topics.