Improved Blow-Up Criterion in a Variational Framework for Nonlinear SPDEs

TL;DR

This paper improves the criteria for solution blow-up in stochastic partial differential equations by extending the blow-up condition to larger variational spaces, ensuring higher regularity of solutions over their lifespan.

Contribution

It introduces an enhanced blow-up criterion within a variational framework, broadening the understanding of solution existence and regularity for nonlinear SPDEs.

Findings

Solutions exist until blow-up in larger variational spaces.

Higher order regularity of initial conditions is preserved in 2D and 3D stochastic Navier-Stokes equations.

The new criterion extends previous results on maximal solutions.

Abstract

We extend recent existence and uniqueness results for maximal solutions of SPDEs through an improved blow-up criterion. Whilst the maximal time of existence is typically characterised by blow-up in the energy norm of solutions, we show instead that solutions exist until blow-up in the larger spaces of the variational framework. The result is applied to show that solutions of 2D and 3D Stochastic Navier-Stokes Equations retain the higher order regularity of the initial condition on their time of existence.