海洋动力学中的边界层问题

Vanishing viscosity limit and boundary layer problems for fluid dynamic equations are always extremely challenging problems in fluid dynamics and applied mathematics. This project will study some vanishing viscosity limit and boundary layer problems for Boussinesq equations with partial viscosity and physical boundaries. More precisely, we will firstly study the vanishing viscosity limit behaviour and boundary layer problem of Boussinesq equations with only horizontal viscosity for the cases with/without diffusion dissipation, for which there are weak layers in the velocity field near the boundary. Secondly, we will study the vanishing viscosity limit and boundary layer problem of Boussinesq equations with only vertical viscosity for the cases with/without diffusion dissipation. In this part, the celebrated Prandtl boundary layer will appear near the boundary in the velocity filed. Meanwhile, this project will also investigate the effect of diffusion dissipation in the stability analysis of boundary layer expansion for Boussinesq equations with partial viscosity. The less dissipation in horizontal or vertical direction for Boussinesq equations will lead to some serious difficulties in the stability analysis of boundary layer expansion, which will require to develop some new methods to overcome these difficulties.

流体力学方程组的粘性消失极限问题与边界层问题一直以来是流体力学与应用数学中极具挑战的难题。本项目将研究一类带物理边界的部分粘性的海洋动力学Boussinesq方程组的粘性消失极限与边界层问题。具体而言，首先，研究只有水平粘性的Boussinesq方程在有扩散耗散和无扩散耗散两种情形的粘性消失极限和边界层问题，速度场在边界附近有弱边界层产生；其次，研究只有垂直粘性的Boussinesq方程在有扩散耗散和无扩散耗散两种情形的粘性消失极限和边界层问题，速度场在边界附近有著名的Prandtl边界层产生；同时，本项目研究扩散耗散在边界层展开的稳定性分析中的作用。耗散项在某些方向上缺失会对Boussinesq方程边界层展开的稳定性分析带来本质的影响，需要发展一些新的方法解决该问题。

海洋动力学模型由Boussinesq方程描述。带物理边界的Boussineq方程的粘性消失极限和边界层问题是具有实际意义和挑战的前沿问题。本项目研究了二维的无扩散的只有垂直粘性的Boussineq方程在半平面满足Navier边界条件时的粘性消失极限问题，证明了存在一致时间使得部分粘性系统的解强收敛到无粘性系统的解。这一结果可以推广到半平面只有水平耗散的Boussinesq系统。我们的研究方法对进一步研究Boussinesq系统满足非滑移边界条件时Prandtl边界层情形有启发作用，明确了研究的关键挑战。. 除了固定边界问题，在本项目的资助下，我们也研究了一类流体自由边界问题解的存在性和渐近行为，包括可压Navier-Stokes-Poisson方程以及带阻尼项和硬核的Euler方程自由边界问题解的整体存在性和平衡态的稳定性，这也是天文物理中的重要问题。