Anticipating Stochastic 2D Navier-Stokes Equations

TL;DR

This paper proves the existence of solutions to the 2D stochastic Navier-Stokes equations with anticipative initial conditions using Malliavin calculus, advancing understanding of stochastic fluid dynamics with complex noise interactions.

Contribution

It introduces a method to establish solutions for SNSE with anticipative initial data employing Malliavin calculus techniques, which is novel in this context.

Findings

Existence of solutions under Malliavin regularity

Handling anticipative initial conditions in SNSE

Application of anticipating calculus techniques

Abstract

In this article, we consider the two-dimensional stochastic Navier-Stokes equation (SNSE) on a smooth bounded domain, driven by affine-linear multiplicative white noise and with random initial conditions and Dirichlet boundary conditions. The random initial condition is allowed to anticipate the forcing noise. Our main objective is to prove the existence of a solution to the SNSE under sufficient Malliavin regularity of the initial condition. To this end we employ anticipating calculus techniques.