A Lower Bound Estimate of Life Span of Solutions to Stochastic 3D Navier-Stokes Equations with Convolution Type Noise

TL;DR

This paper establishes a lower bound estimate for the lifespan of solutions to stochastic 3D Navier-Stokes equations with convolution-type noise, using a fixed point approach without requiring regular initial data.

Contribution

It provides the first lower bound estimates for solution lifespan in stochastic 3D Navier-Stokes equations with convolution noise, transforming the problem into random PDEs.

Findings

Lower bound estimates for solution lifespan obtained

Fixed point method successfully applied to stochastic PDEs

Results applicable to irregular initial data

Abstract

In this paper we investigate the stochastic 3D Navier-Stokes equations perturbed by linear multiplicative Gaussian noise of convolution type by transformation to random PDEs. We are not interested in the regularity of the initial data. We focus on obtaining bounds from below for the life span associated with regular initial data. The key point of the proof is the fixed point argument.