High Order Smoothness for Stochastic Navier-Stokes Equations with Transport and Stretching Noise on Bounded Domains

TL;DR

This paper establishes high-order regularity results for stochastic 2D Navier-Stokes equations with transport and stretching noise on bounded domains, achieving strong solutions with or without hyperdissipation.

Contribution

It introduces new energy estimates in high-order Sobolev spaces for stochastic Navier-Stokes equations with boundary conditions, enabling strong solutions and regularity results.

Findings

Achieved energy estimates in high-order Sobolev spaces for stochastic Navier-Stokes.

Constructed strong solutions with hyperdissipation for any smoothness order.

Obtained smoothness results without hyperdissipation on the torus in 2D and 3D.

Abstract

We obtain energy estimates for a transport and stretching noise under Leray Projection on a 2D bounded convex domain, in Sobolev Spaces of arbitrarily high order. The estimates are taken in equivalent inner products, defined through powers of the Stokes Operator with a specific choice of Navier boundary conditions. We exploit fine properties of the noise in relation to the Stokes Operator to achieve cancellation of derivatives in the presence of the Leray Projector. As a result, we achieve an additional degree of regularity in the corresponding Stochastic Navier-Stokes Equation to attain a true strong solution of the original Stratonovich equation. Furthermore for any order of smoothness, we can construct a strong solution of a hyperdissipative version of the Stochastic Navier-Stokes Equation with the given regularity; hyperdissipation is only required to control the nonlinear term in the presence of a boundary. We supplement the result by obtaining smoothness without hyperdissipation on the torus, in 2D and 3D on the lifetime of solutions.