Fluid Dynamics of Geometric Rough Flows

In the PIs previous work Variational principles for fluid dynamics on rough paths , (attached to final report), they explain how fluid models on geometric rough paths can be used in the context of stochastic parameterization schemes and uncertainty quantification. They have produced enhanced capabilities for quantifying uncertainty in fluid dynamics using the fundamental theorems of rough-path mathematics. These enhanced capabilities comprise the rough Lie chain rule and variational principles for fluid mechanics on geometric rough paths. These capabilities generate further opportunities to analyze the uncertainty of fluid systems. In their most recent work, Solution properties of the incompressible Euler system with rough path advection (attached to final report), they establish the well-posedness of the rough Euler equations and characterized solutions as critical points of the Hamilton-Pontryagin and Clebsh variational principles. This work provides a bridge between two previously unrelated areas of Mathematics; namely, Geophysical Fluid Dynamics and Rough Path Theory. The research team believe crossing this accomplishment could be highly beneficial for both areas. On one hand it will offer sound theoretical support for developing rough-path models in applications such as Numerical Weather Prediction, Data Assimilation, Ocean Dynamics, Atmospheric Science, perhaps even Turbulence. These rough-path models will benefit from a plethora of available theoretical results (stability results, support theorems, large deviation principles, splitting schemes) resulting from their characterization as random dynamical systems generated by the solutions of the rough partial differential equations. In turn, the new connection between GFD and GRPs can be expected to become a fruitful source of open problems in mathematics.