随机偏微分方程的Schauder理论及其应用

The proposed program will study the solvability and regularity problem for stochastic parabolic partial differential equations. Based on some proper Hölder spaces, the program aims to develop a new method of pointwise estimates, one the one hand, to prove the interior, boundary and global Schauder estimates for the initial-boundary value problem for stochastic parabolic equations and then obtain the existence and uniqueness of solutions to such equations in Hölder spaces, and on the other hand, to study the partial regularity problem for stochastic parabolic equations, which is to find what regularity the solutions possess if the free terms of the equations are Hölder continuous only with respect to partial variables. The program will also investigate the applications of the Schauder theory to numerical and nonlinear problems of stochastic parabolic equations.

本项目拟研究随机抛物型偏微分方程的可解性和正则性问题。以恰当的Hölder型空间为基础，本项目旨在发展一套新的逐点估计方法，一方面证明随机抛物型方程初边值问题的内部、边界和全局Schauder估计，籍此获得方程在Hölder型空间中的解的存在唯一性，另一方面研究随机抛物型方程的部分正则性问题，即当方程非齐次项仅关于部分变量Hölder连续时，方程的解具有何种正则性。本项目还将初步探讨所获Schauder理论在随机抛物型方程数值计算和非线性问题中的应用。

本项目建立了二阶随机偏微分方程的Schauder估计，并推广至二阶随机偏微分方程组和高阶随机偏微分方程；对于后两者，给出了L^p估计所需的修正的随机抛物型条件，并通过实例说明了这些条件的最优性。项目给出了二阶随机抛物型方程Dirichlet问题存在整体正则解的最优相容性条件。上述成果解决了随机偏微分方程领域的数个公开问题。项目还获得了计划之外的若干成果，为后续的研究做好准备。