两类随机偏微分方程的适定性和动力学性质

Stochastic partial differential equation has been one of most popular topics in mathematical research in the last few years. It is used to depict the phenomena and models with disturbance, uncertainties in the real world, and also has wide applications in many research fields such as physics, chemistry, biology, climate and so on. This project aims to study two types of stochastic partial differential equations----stochastic Camassa-Holm equation (CH equation for short) and stochastic Kuramoto-Sivashinsky and Ginzburg-Landau equations (KS-GL equations for short). Combining the techniques and methods of stochastic analysis with the PDE theories, we will study three specific issues which include: 1.the Gibbs measure and the existence of global solution of stochastic higher order modified CH equation; 2.the well-posedness of solution and the large deviation of stochastic higher order modified CH equation driven by Lévy process; 3.the random attractors of the stochastic KS-GL Equations perturbed by additive noises. These three issues are related to the study of well-posedness and dynamical properties of stochastic CH equation and stochastic KS-GL equations, which has applications to mathematical physical models with random disturbance in practical background.

随机偏微分方程近年来成为数学研究的热点之一，它被用以刻画现实世界中受噪声干扰的、具有不确定性的现象和模型，并被广泛地应用于物理、化学、生物、气象学等各领域。本项目以流体动力学中的两类重要的随机偏微分方程----随机Camassa-Holm方程（简记为CH方程）和随机Kuramoto-Sivashinsky和Ginzburg-Landau方程组（简记为KS-GL方程组）为研究对象，利用随机分析的技巧和方法，结合PDE理论对三个具体问题开展研究，包括：1.随机高阶修正CH方程的Gibbs测度及方程整体解的存在性；2.Lévy过程驱动下的高阶修正CH方程的适定性和大偏差理论；3.随机KS-GL方程组的随机吸引子。这三个问题涉及到对随机CH方程和随机KS-GL方程组的适定性和动力学性质等方面的研究，其研究结果可应用于具有实际背景的随机干扰下的数学物理模型。

本项目利用随机分析的技巧与方法，结合偏微分方程的理论，研究了流体力学中几类非线性的随机偏微分方程。项目的研究内容涉及随机偏微分方程的适定性、正则性、Gibbs测度等理论，可应用到具有实际背景的随机干扰下的随机数学模型，具有理论和应用价值。.本项目顺利完成既定研究计划，研究了随机化初值且周期边界的随机高阶修正Camassa-Holm方程（简记为CH方程）的Gibbs测度，并利用Gibbs测度证明此类方程整体解的存在性；验证了随机Kuramoto-Sivashinsky和Ginzburg-Landau 方程组（简记为KS-GL方程组）的解生成一个连续的随机动力系统，进一步得到了该方程组的随机吸引子。在完成原有研究计划的基础上，对一些相关问题进行了扩展研究，证明了非Markov系数的随机微分方程与系数平方增长的倒向随机偏微分方程的联系和应用，推广了Truman-王凤雨-吴奖伦2012年Markov系数的随机微分方程与偏微分方程的联系的结果。.在项目执行期间，项目负责人共计发表论文5篇，发表的杂志包括“Chinese Annals of Mathematics, Series B”、“Acta Mathematica Sinica, English Series”、“Applied Mathematics Letters”等学术期刊。项目的研究结果推动了随机高阶修正CH方程、随机KS-GL方程组适定性、动力学性质等研究问题的进展，并将理论研究应用到随机干扰下的其他数学物理模型。