Space-time approximation of local strong solutions to the 3D stochastic Navier-Stokes equations

TL;DR

This paper establishes optimal convergence rates for space-time discretisation of local strong solutions to the 3D stochastic Navier-Stokes equations, accounting for potential blow-up scenarios.

Contribution

It introduces a novel approach using discrete stopping times to analyze convergence rates of discretised solutions to the stochastic Navier-Stokes equations.

Findings

Convergence of order 1 in space

Convergence of order up to 1/2 in time

Results valid up to possible solution blow-up

Abstract

We consider the 3D stochastic Navier-Stokes equation on the torus. Our main result concerns the temporal and spatio-temporal discretisation of a local strong pathwise solution. We prove optimal convergence rates in for the energy error with respect to convergence in probability, that is convergence of order 1 in space and of order (up to) 1/2 in time. The result holds up to the possible blow-up of the (time-discrete) solution. Our approach is based on discrete stopping times for the (time-discrete) solution.