Analytical Properties for a Stochastic Rotating Shallow Water Model under Location Uncertainty

TL;DR

This paper introduces a stochastic version of the rotating shallow water model incorporating location uncertainty, proves existence and uniqueness of solutions, and constructs a global weak solution using finite-dimensional approximations.

Contribution

It develops a novel stochastic formulation of the rotating shallow water model with proven well-posedness and solution existence, advancing the mathematical understanding of uncertain geophysical fluid models.

Findings

Proved the stochastic model has unique maximal strong solutions.

Constructed a global weak solution for the stochastic PDE.

Used finite-dimensional Littlewood-Paley space for approximation.

Abstract

The rotating shallow water model is a simplification of oceanic and atmospheric general circulation models that are used in many applications such as surge prediction, tsunami tracking and ocean modelling. In this paper we introduce a class of rotating shallow water models which are stochastically perturbed in order to incorporate model uncertainty into the underlying system. The stochasticity is chosen in a judicious way, by following the principles of location uncertainty, as introduced in [M\'emin, 2014]. We prove that the resulting equation is part of a class of stochastic partial differential equations that have unique maximal strong solutions. The methodology is based on the construction of an approximating sequence of models taking value in an appropriately chosen finite-dimensional Littlewood-Paley space. Finally, we show that a distinguished element of this class of stochastic partial differential equations has a global weak solution.