On the well-posedness of stochastic Boussinesq equations with cylindrical multiplicative noise

TL;DR

This paper proves local well-posedness for stochastic Boussinesq equations with cylindrical multiplicative noise, preserving geometric properties and providing criteria for blow-up and estimates for similar SPDEs.

Contribution

It introduces a novel stochastic formulation of Boussinesq equations that maintains geometric properties and establishes well-posedness and blow-up criteria.

Findings

Proved local well-posedness of stochastic Boussinesq equations.

Constructed a blow-up criterion for solutions.

Derived general estimates for similar SPDEs.

Abstract

The Boussinesq equations are fundamental in meteorology. Among other aspects, they aim to model the process of front formation. We use the approach presented in [Hol15] to introduce stochasticity into the incompressible Boussinesq equations. This is, we introduce cylindrical transport noise in a way that the geometric properties in the Euler-Poincar\'{e} formulation are preserved. One of our main results establishes the local well-posedness of regular solutions for these new stochastic Boussinesq equations. We also construct a blow-up criterion and derive some general estimates, which are crucial for showing well-posedness of a wide range of similar SPDEs.