An Essay On Well-Posedness Of A Few Kinds Of Stochastic Partial Differential Equations And Related Problems

Partial differential equations can be as a central tool in the description of natural phenomenons, and as the principal mode of analytical study of models in the physical science, and thus more and more people concentrate their attention on studying them. However, there is always a certain gap between the results deriving from the models given by partial differential equations and the reality in some physical phenomenons, such as atmospheric physics. Therefore, people want to obtain reasonable fruits by adding logical stochastic perturbations. Based upon this, we plan to unfold our dis-cussion on several kinds of stochastic partial differential equations, and establish the fundamental results of existence and uniqueness on weak solutions.We divide our present Ph.D. thesis into six chapters.Chapter1aims to introduce the historical background, the current situation and the main results of our thesis.In Chapter2, we outline some celebrate results on the transport equation and con-tinuity equation, and sketch some element notions in probability.Chapter3is devoted to study the existence and uniqueness of weak Lp-solutions for the Fokker-Planck equation. Firstly, we derive a random Liouville theorem from stochastic differential equations, which is analogue of the Liouville theorem in differ-ential equations, and further introduce the the Fokker-Planck equation. Next the renor-malization technique and the existing transport theory apply, we conclude the existence for weak solutions. Later on, by regularization, one obtains the uniqueness. After that, we take Fokker-Planck-Boltzmann equation as a typical example to illustrate our result. Finally, one establishes the existence as well as uniqueness for weak solutions to the fractional Fokker-Planck equation. All the results are new.Chapter4is intended to argue the existence and uniqueness of weak solutions for the stochastic Ginzburg-Landau equation. Initially, the equation can be changed into a nonlinear transport equation, thus we gain the weak solutions by making use of the nonlinear transport theory. We extend and improve part known results.Chapter5is then devoted to investigate the well-posedness of weak solutions for the nonlinear hyperbolic balance laws by nonlinear stochastic perturbations. With the aid of a kinetic formulation, one shows the solvability of the stochastic hyperbolic balance law is equivalent to the solvability of a non-homogeneous linear stochastic transport equation, then by regularizing procedure, we maintain the uniqueness of weak solutions. Last, by the relation, the existence of weak solutions is derived mutatis mutandis, and the results are new.In the final Chapter6, we summarize our works and draw the future studies.