Global well-posedness of 2D Navier-Stokes with Dirichlet boundary fractional noise

TL;DR

This paper establishes the global well-posedness and interior regularity for the 2D Navier-Stokes equations influenced by fractional noise at the boundary, modeling a stochastic oceanic flow with fractional temporal characteristics.

Contribution

It introduces a novel stochastic boundary condition with fractional noise for 2D Navier-Stokes, proving well-posedness and regularity results in this context.

Findings

Proved global well-posedness of 2D Navier-Stokes with fractional boundary noise.

Established interior regularity under stochastic boundary conditions.

Modeled oceanic flow with fractional Gaussian noise at the boundary.

Abstract

In this paper, we prove the global well-posedness and interior regularity for the 2D Navier-Stokes equations driven by a fractional noise acting as an inhomogeneous Dirichlet-type boundary condition. The model describes a vertical slice of the ocean with a relative motion between the two surfaces and can be thought of as a stochastic variant of the Couette flow. The relative motion of the surfaces is modeled by a Gaussian noise which is coloured in space and fractional in time with Hurst parameter greater than 3/4.