Existence and Uniqueness of Maximal Solutions to SPDEs with Applications to Viscous Fluid Equations

TL;DR

This paper establishes criteria for the existence and uniqueness of maximal solutions to SPDEs, with detailed theoretical results and applications to stochastic fluid dynamics models like the 3D SALT Navier-Stokes equations.

Contribution

It provides new abstract well-posedness criteria for SPDEs and applies these to complex viscous fluid equations with various noise types.

Findings

Criteria for unique maximal solutions to SPDEs are established.

The results are applied to the 3D SALT Navier-Stokes equations.

The paper details conditions under which solutions exist and are unique.

Abstract

We present two criteria to conclude that a stochastic partial differential equation (SPDE) posseses a unique maximal strong solution. This paper provides the full details of the abstract well-posedness results first given in arXiv:2202.09242v2, and partners a paper which rigorously addresses applications to the 3D SALT (Stochastic Advection by Lie Transport) Navier-Stokes Equation in velocity and vorticity form, on the torus and the bounded domain respectively. Each criterion has its corresponding set of assumptions and can be applied to viscous fluid equations with additive, multiplicative or a general transport type noise.