Fractal dimension of singular times for SPDEs: Energy bounds, criticality, and weak-strong uniqueness

TL;DR

This paper investigates the fractal dimension of singular times in solutions to SPDEs, establishing bounds and measure properties, and applies the results to 3D Navier-Stokes equations with stochastic noise.

Contribution

It introduces a general framework for analyzing the fractal dimension of singular times in SPDEs, extending classical results to stochastic settings and providing new regularity insights.

Findings

Sets of singular times have fractal dimension at most 1 - ℓ * Exc.

The (1 - ℓ * Exc)-dimensional measure of singular times is zero.

Extension of classical bounds on singular times to stochastic Navier-Stokes equations.

Abstract

For several physically relevant SPDEs, it is known that global weak solutions coexist with local strong ones. Typically, weak-strong uniqueness results are known, and ensure that the global and strong solutions coincide as long as the latter exist. Times at which a weak solution does not coincide with a strong one are called singular times. Determining their fractal dimension is fundamental to capturing the regularity of weak solutions. We define singular times for a wide class of semilinear SPDEs. We show that sets of singular times have fractal dimension (i.e., Hausdorff and/or Minkowski) at most $ 1-\ell\, \mathsf{Exc}$, where $\ell$ and $\mathsf{Exc}$ are the time integrability and the excess of spatial regularity compared to the critical regularity of the energy bound associated with weak solutions, respectively. Moreover, their corresponding $(1-\ell\,\mathsf{Exc} )$-dimensional measure is zero. We formulate and apply our theory to quenched strong Leray-Hopf solutions of 3D Navier-Stokes equations (NSEs) with physically relevant noises, including rough Kraichnan and Lie transport. In particular, we extend the fundamental $1/2$-dimensional bound of Leray and Scheffer on singular times for 3D NSEs to the stochastic setting, and we prove new conditional results under supercritical Serrin's conditions, irrespective of the roughness of the noise. Our framework is new even in the deterministic case, and provides the first partial regularity results for weak solutions to SPDEs with multiplicative noise.