具有分数阶耗散的Boussinesq方程组相关模型解的性态研究

In hydrodynamics, Boussinesq equation is a low-dimensional model in fluid dynamics, which plays a very important role in the study of Raleigh-Bernard convection. This project focuses on the releted models on Boussinesq equations with fractional dissipative. This research priorities include: the behavior of solutions and boundary layer problem of different fractional Boussinesq equation with different boundary conditions. On the basis of studying the fractional Schrödinger equation, we discuss the invariant solutions and their properties, conservation laws of the fractional Boussinesq-Schrödinger equations. Based on studying the global well-posedness and long-time behavior of special structural solutions for high-dimensional NSB equations and the regularity of solution in different spaces, we finally study the well-posedness, regularity and asymptotic behaviors of the solution of high-dimensional fractional (partial fractional) NSB equations. This project will not only further improve the theoretical system of fractional nolinear evolution equations, but also provide support for the innovation on research methods and the related properties of fluid dynamics models.

水动力学中，Boussinesq方程组是水动力方程的一个低维模型，在Raleigh-Bernard对流的研究中起着重要的作用。本项目的研究对象为具有较大难度的分数阶Boussinesq方程组相关模型，研究重点包括：对具有不同边界条件的分数阶Boussinesq方程组解的性态以及边界层问题进行研究；在研究分数阶Schrödinger方程的基础上，讨论分数阶Boussinesq-Schrödinger方程组的不变解及其性质、守恒律问题；以研究高维Navier-Stokes-Boussinesq(NSB)方程组具有特殊结构解的整体适定性及长时间行为、解在不同空间中的正则性理论为前提，最终深入讨论高维分数阶(部分分数阶)NSB方程组的适定性、正则性及渐近性。本项目不仅进一步完善了分数阶非线性发展方程研究的理论体系，也为更广泛意义下的流体动力学模型相关属性的研究方法创新提供支持。

水动力学中，Boussinesq方程组是水动力方程的一个低维模型，在Raleigh-Bernard对流的研究中起着重要的作用。本项目主要围绕分数阶Boussinesq方程、Navier-Stokes、Schrödinger方程及多种耦合方程完成了系列成果，在初边值问题，球对称解的全局存在性、唯一性和渐近性态，强解的全局存在性、唯一性和指数稳定性等方面取得阶段性成果。通过开展本项目，不仅进一步完善了分数阶非线性发展方程研究的理论体系，也为更广泛意义下的流体动力学模型相关属性的研究方法创新提供支持。