Interaction of Deterministic or Stochastic Forces on Flat or Moving Interfaces

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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.