Interaction of Deterministic or Stochastic Forces on Flat or Moving Interfaces

这个项目涉及地球物理流动与确定或随机力在平坦或移动边界上的相互作用。在Sobolev空间和连续函数中，由具有平坦边界和齐次边界数据的几何图形中的原始方程控制的地球物理流的演化是相当容易理解的。事实上，当考虑在固定的平坦边界和没有力时，已知底层方程是强全局适定的。当应用确定性或随机力或当底层域被允许为移动界面（如在移动或自由边值问题的情况下）时，这种情况不再发生。本项目旨在通过研究随机强迫通过输运噪声的影响，研究由Wiener过程描述的风驱动边界条件，构建相关自由边值问题的强解，以及研究确定性设置下的流固相互作用或流海冰耦合，同时推进对外力和运动界面的更深入理解。更详细地说，我们将：-研究受输移噪声影响的非等温原始方程的全局存在性结果，-研究原始方程的自由边值问题，-研究受随机风驱动边界条件影响的原始方程的全局强解，并将设置扩展到自由表面，-考虑关于全局解和自由界面的耦合大气-海洋-海冰模型。-研究地球物理流动和冰山等冰结构的流体-结构相互作用。相对于Navier-Stokes方程，在地球物理流动的背景下研究输运噪声是一个相当新的领域。起点是经典的布辛尼斯克近似和流体静力近似的随机对应。利用负阶Sobolev空间中的能量估计，研究了随机风驱动边界条件下原始方程的局部路径解推广到全局解的问题。在关于自由或移动界面的确定性设置中，我们计划将问题转换为固定域上的方程组，并应用最新的解耦方法。在平面或自由界面上的大气-海洋-海冰耦合模式的分析将依赖于这些方法。

This project concerns the interaction of geophysical flows with deterministic or stochastic forces on flat or moving boundaries. The evolution of a geophysical flow governed by the primitive equations in geometries with flat boundaries and homogeneous boundary data is rather well understood in the setting of Sobolev spaces as well as for continuous functions. In fact, the underlying equations are known to be strongly globally well-posed, when considered on fixed flat boundaries and without forces. This is no longer the case when deterministic or stochastic forces are applied or when the underlying domain is allowed to be a moving interface as in the case of moving or free boundary value problems. This project aims at a simultaneous advance in a deeper understanding of outer forces and moving interfaces by investigating four circles of problems: study the effects of stochastic forcing through transport noise, investigate wind driven boundary condition described by a Wiener process, construct strong solutions to associated free boundary value problems and study fluid-structure interaction or fluid-sea ice coupling within the deterministic setting. In more detail we shall: - study global existence results for non-isothermal primitive equations subject to transport noise, - investigate the free boundary value problem for the primitive equations, - examine global strong solutions for the primitive equations subject to stochastic wind driven boundary conditions and extend the setting to free surfaces, - consider coupled atmosphere-sea ice-ocean models with regard to global solutions and free interfaces, - investigate fluid-structure interactions for geophysical flows and ice structures such as icebergs. In contrast to the Navier-Stokes equations, the study of transport noise in the context of geophysical flows is a rather new field. The starting points are the stochastic counterparts of the classical Boussinesq and hydrostatic approximations. The question on the extension of the local pathwise solution to the primitive equations subject to stochastic wind driven boundary conditions to a global one will be investigated by energy estimates in Sobolev spaces of negative order. In the deterministic setting concerning free or moving interfaces we plan to transform the problems to sets of equations on fixed domains and apply recent decoupling methods. The analysis of the coupled atmosphere-sea ice-ocean model on flat or free interfaces will rely on these methods.