Abstract
We study the two dimensional primitive equations in the presence of multiplicative stochastic forcing. We prove the existence and uniqueness of solutions in a fixed probability space. The proof is based on finite dimensional approximations, anisotropic Sobolev estimates, and weak convergence methods.
Highlights
The Primitive equations are a ubiquitous model in the study of geophysical fluid dynamics
They can be derived from the compressible NavierStokes equations by taking advantage of various properties common to geophysical flows
In the works of Mikulevicius and Rozovsky [26] and of Brzezniak and Peszat [5] the case of arbitrary space dimensions for local solutions evolving in Sobolev spaces of type W 1,p for p > d is addressed. It is with this background in mind that we present the following examination of the two dimensional Primitive equations in the presence of multiplicative white noise terms on a preordained probability space
Summary
The Primitive equations are a ubiquitous model in the study of geophysical fluid dynamics. In the first section we introduce the model, providing an overview of the relevant function spaces and establish some anisotropic Sobolev type estimates on the nonlinear terms of the equation. We say that an Ft adapted process u is a weak-strong solution to the stochastically forced primitive equation if: u ∈ C([0, T ]; H) a.s. u ∈ Lp(Ω; L∞([0, T ]; H) ∩ L2([0, T ]; V ))
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