非线性双曲型随机偏微分方程及其相关研究

The present project aims at studying on nonlinear stochastic partial differential equations of hyperbolic type arising in physics, engineering, mechanics of materials and aeroelasticity. The results we obtained about this model are applicable to a wide class of equations including nonlinear stochastic wave equations and nonlinear stochastic beam equations subject to either periodic boundary conditions, or the clamped boundary conditions, or the hinged boundary conditions. We will prove the existence and uniqueness of global mild solutions to stochastic differential equations of hyperbolic type driven by Lévy noises. We will also prove the ultimate boundedness, asymptotic stability of the zero solution and the existence of invariant measure. Moreover, the maximal inequalities in infinite dimensional spaces will be investigated in this project. We will extend the maximal inequalities for stochastic convolutions to a more general Banach space setting, which will include Sobolev spaces. The completion of the proposed project will enrich the existing theoretical achievements on stochastic partial differential equations and stochastic analysis in infinite dimensional spaces, and the approach we have developed can provide a complementary perspective to the study of stochastic differential equations of various types.

本项目将以物理、工程、材料力学及气动力弹性学中的一类非线性双曲型随机偏微分方程为主要研究对象。我们建立的模型不仅可应用于非线性随机波动方程，还可应用于周期边界条件、固定支承边界条件或铰直承边界条件下的非线性随机梁方程。我们将证明和解决由Lévy噪音驱动的双曲型随机偏微分方程的全局mild解的存在唯一性。并在此基础上进一步证明解的一致有界性、渐进稳定性和不变测度的存在性等问题。本项目也将研究在无穷维空间下随机卷积的极大值不等式。我们计划将随机卷积的极大值不等式推广到Banach空间下，从而可应用于Sobolev空间。我们的研究成果可以丰富随机偏微分方程、无穷维空间上随机分析的基础理论内容，同时我们在研究方法上的突破也将对不同类型方程之间的研究方法给予启示。

受机械工程和气动力弹性学上一类挑战性问题的启发，我们在该项目中研究了一类由Lévy噪音驱动的双曲型随机偏微分方程。我们证明得到了无穷维上关于局部mild解的伊藤公式。该伊藤公式对于证明我们的主要结果至为关键。在给定的条件下，我们证明了这类方程全局解的存在唯一性，同时建立了解的渐近稳定性和一致有界性。另外在基本的假设条件下，我们还证明了本项目所研究的双曲型随机偏微方程框架和结果可以应用于非线性随机波动方程、周期边界条件下的非线性随机梁方程、固定支承边界条件或铰支承边界条件下的非线性随机梁方程等。Burkholder-Davis-Gundy不等式和随机卷积的极大值不等式是证明方程解的存在唯一性和解的正则性等性质的关键工具。本项目还证明了几类Burkholder-Davis-Gundy不等式并得到了Banach空间上2个伊藤公式。同时我们利用伊藤公式证明了Banach空间上由Lévy型噪音驱动的随机卷积的极大值不等式和指数尾估计。作为应用，我们证明了Lévy型噪音驱动的随机quasi-geostrophic方程mild解的存在唯一性。