Well-Posedness of the 3D Stochastic Primitive Equations with Transport Noise

TL;DR

This paper proves the well-posedness of the 3D stochastic primitive equations with transport noise, establishing local and global solutions under certain boundary conditions and noise growth assumptions.

Contribution

It introduces new results on the existence and uniqueness of solutions for the stochastic 3D primitive equations with gradient-dependent noise and specific boundary conditions.

Findings

Unique maximal strong solutions established for the equations.

Global existence proved under certain boundary conditions and noise assumptions.

An explicit example of infinite-dimensional noise depending on vertical averages provided.

Abstract

We show that that the stochastic 3D primitive equations with either the physical boundary conditions or Neumann boundary conditions on the top and bottom and Dirichlet boundary condition on the sides driven by multiplicative gradient-dependent white noise have unique maximal strong solutions both in stochastic and PDE sense under certain assumptions on the growth of the noise. For the latter boundary conditions global existence is established using an argument based on decomposition of vertical velocity to barotropic and baroclinic modes and an iterated stopping time argument. An explicit example of non-trivial infinite dimensional noise depending on the vertical average of the horizontal gradient of horizontal velocity is presented.