倒向随机微分发展系统及其应用
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摘要
本文主要讨论倒向随机微分发展系统及其应用.
第一章考察一般的倒向重随机微分发展系统,建立关于Banach空间取值的倒向随机微分发展系统的Ito公式,利用基于单调算子理论的弱收敛方法证明解的存在唯一性.作为应用,给出拟线性倒向重随机偏微分方程弱解的存在唯一性定理,为后面的第四章做准备.
第二章建立耦合的有限维正倒向随机微分系统与粘性不可压缩Navier-Stokes方程之间的联系,推广Feynman-Kac公式,为Navier-Stokes方程的解给出概率表示.而且,利用概率分析的方法证明这类正倒向随机微分系统局部解的存在唯一性；对于两维情形和小雷诺数情形,还进一步证明全局解的存在唯一性.
第三章给出全空间上半线性超抛物型倒向随机偏微分方程的Lp理论及其比较定理.
第四章中首先证明带有Sobolev空间系数的有界区域上拟线性倒向随机偏微分方程Dirichlet问题弱解的存在唯一性,然后使用De Giorgi迭代的技术证明其弱解满足最大值原理和局部最大模估计.
最后,第五章考察2维倒向随机Navier-Stokes方程,在空间周期性边界约束条件下证明其强解的存在唯一性.
The present dissertation is concerned with backward stochastic differential evolutionary systems and their applications.
Chapter1is concerned with general backward doubly stochastic differential evolutionary systems. An Ito formula is established for the Banach space-valued backward doubly stochastic differential evolutionary systems. Using the weak convergence method on basis of monotone operator theory, the existence and uniqueness of the solution is proved. As an application, the existence and uniqueness is studied on the weak solution for quasilinear backward doubly stochastic PDEs, which will be used in Chapter4.
In Chapter2, the connections are addressed between a class of coupled forward-backward stochastic differential systems and the viscous incompressible Navier-Stokes equations, which generalizes the Feynman-Kac formula and gives a probabilistic representation for the solutions of Navier-Stokes equations. Moreover, a self-contained probabilistic proof is given for the existence and uniqueness of the local Hm-solution, and the local solution can be extended uniquely to be a global one for both cases of dimension2and small Reynolds number.
In Chapter3, the LP theory and a comparison theorem are established for the semilinear super-parabolic backward stochastic PDEs on the whole space.
In Chapter4, the existence and uniqueness is first shown for the weak solution for the Dirichlet problem of the quasilinear backward stochastic PDEs with Sobolev coefficients on bounded domains. Then the De Giorgi iteration scheme is used to prove the maximum princi-ples and the local maximum estimates.
Finally in Chapter5,2-dimensional backward stochastic Navier-Stokes equation with nonlinear external forcing is investigated. The existence and uniqueness is shown for the strong solution under the spatially periodic boundary condition.
第一章考察一般的倒向重随机微分发展系统,建立关于Banach空间取值的倒向随机微分发展系统的Ito公式,利用基于单调算子理论的弱收敛方法证明解的存在唯一性.作为应用,给出拟线性倒向重随机偏微分方程弱解的存在唯一性定理,为后面的第四章做准备.
第二章建立耦合的有限维正倒向随机微分系统与粘性不可压缩Navier-Stokes方程之间的联系,推广Feynman-Kac公式,为Navier-Stokes方程的解给出概率表示.而且,利用概率分析的方法证明这类正倒向随机微分系统局部解的存在唯一性；对于两维情形和小雷诺数情形,还进一步证明全局解的存在唯一性.
第三章给出全空间上半线性超抛物型倒向随机偏微分方程的Lp理论及其比较定理.
第四章中首先证明带有Sobolev空间系数的有界区域上拟线性倒向随机偏微分方程Dirichlet问题弱解的存在唯一性,然后使用De Giorgi迭代的技术证明其弱解满足最大值原理和局部最大模估计.
最后,第五章考察2维倒向随机Navier-Stokes方程,在空间周期性边界约束条件下证明其强解的存在唯一性.
The present dissertation is concerned with backward stochastic differential evolutionary systems and their applications.
Chapter1is concerned with general backward doubly stochastic differential evolutionary systems. An Ito formula is established for the Banach space-valued backward doubly stochastic differential evolutionary systems. Using the weak convergence method on basis of monotone operator theory, the existence and uniqueness of the solution is proved. As an application, the existence and uniqueness is studied on the weak solution for quasilinear backward doubly stochastic PDEs, which will be used in Chapter4.
In Chapter2, the connections are addressed between a class of coupled forward-backward stochastic differential systems and the viscous incompressible Navier-Stokes equations, which generalizes the Feynman-Kac formula and gives a probabilistic representation for the solutions of Navier-Stokes equations. Moreover, a self-contained probabilistic proof is given for the existence and uniqueness of the local Hm-solution, and the local solution can be extended uniquely to be a global one for both cases of dimension2and small Reynolds number.
In Chapter3, the LP theory and a comparison theorem are established for the semilinear super-parabolic backward stochastic PDEs on the whole space.
In Chapter4, the existence and uniqueness is first shown for the weak solution for the Dirichlet problem of the quasilinear backward stochastic PDEs with Sobolev coefficients on bounded domains. Then the De Giorgi iteration scheme is used to prove the maximum princi-ples and the local maximum estimates.
Finally in Chapter5,2-dimensional backward stochastic Navier-Stokes equation with nonlinear external forcing is investigated. The existence and uniqueness is shown for the strong solution under the spatially periodic boundary condition.
引文
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