一类非高斯噪声驱动的随机流体方程的动力学研究

In this project, we study the dynamics of stochastic fluid equations driven by a class of non-Gaussian noises. The Navier-Stokes equations were described the equation of viscous Newtonian fluid in fluid mechanics which are very important in fluid mechanics. Taking the stochastic external factors into account, firstly, we study the stochastic three-dimensional Navier-Stokes equations with damping driven by the non-degenerate and degenerate noises, and shall obtain the martingale solution, well-posedness of strong solutions, the existence and uniqueness of the invariant measure and random attractor. Meanwhile, we study the small time large deviation, large deviation and moderate deviation of the solution of the equations by using weak convergence method. Secondly, we study the well-posedness and ergodicity of a class of stochastic three-dimensional Navier-Stokes equations with damping driven by jump noise. Meanwhile, we study the small time asymptotics, large deviation, moderate deviation of the solution of the equations by using weak convergence method. Thirdly, we study a class of stochastic fluid equations, such as, stochastic micropolar equations, stochastic MHD equations and stochastic magneto-micropolar equations, the well-posedness and dynamics of solutions are studied. Finally, we study the well-posedness of a class of stochastic fluid system by constructing some special noises.

本项目拟研究一类非高斯噪声驱动的随机流体方程的动力学。Navier-Stokes方程是流体力学中描述粘性牛顿流体的方程，在流体力学中有十分重要的意义。考虑随机效应的影响，首先非退化与退化噪声驱使的具有阻尼项的随机三维Navier-Stokes方程被研究，拟得出鞅解、强解的适定性、不变测度的存在唯一性和随机吸引子，同时，通过弱收敛方法，研究方程解的小时间大偏差、大偏差、中偏差。第二，建立跳噪声过程驱动的一类具有阻尼项的随机三维Navier-Stokes方程的适定性和遍历性，同时，通过弱收敛方法，研究方程解的小时间渐近行为、大偏差、中偏差。第三，一类随机流体方程被研究。例如，随机Micropolar方程、随机MHD方程和随机Magneto-micropolar方程，拟建立解的适定性与动力学。最后，构造一些特殊噪声，研究一类随机流体方程的适定性。

本项目研究一类非高斯噪声驱动的随机流体方程的动力学。Navier-Stokes方程是流体力学中描述粘性牛顿流体的方程，在流体力学中有十分重要的意义。考虑随机效应的影响，首先非退化与退化噪声驱使的具有阻尼项的随机三维Navier-Stokes方程被证明，得出鞅解、强解的适定性、不变测度的存在唯一性和随机吸引子，同时，通过弱收敛方法，证明方程解的小时间大偏差、大偏差、中偏差。第二，证明了跳噪声过程驱动的一类具有阻尼项的随机三维Navier-Stokes方程的适定性和遍历性，同时，通过弱收敛方法，证明方程解的小时间渐近行为、大偏差、中偏差。第三，证明了一类流体方程的适定性和吸引子，例如，具有阻尼项的三维Micropolar方程、具有阻尼项的三维MHD方程和分数阶Magneto-micropolar方程。最后，证明了一类随机各向异性流体方程的适定性。在项目期间，项目主持人发表SCI论文20多篇，例如，Zeitschrift für Angewandte Mathematik und Physik，Communications in Mathematical Sciences等期刊。