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

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.

Figures (3)

• Figure 1: Derivation of Theorem \ref{['t:NSE_intro']} from Theorem \ref{['t:abstract_intro']}. We used that the critical Sobolev threshold for 3D NSEs is equal to $-1$ (see below \ref{['eq:invariant_maps_NSE']}), and the factor $\frac{1}{2}$ in the excess formula $\mathsf{Exc}$ is because in \ref{['eq:NSE_intro']} the leading operators are of second-order (or in other words, parabolic scaling with time counted as the unit).
• Figure 2: The striped region is the area of applicability of Theorem \ref{['t:abstract_intro']}. Here, $\ell$ is as in \ref{['eq:energy_bound_xx_intro']}, and $\mathsf{Exc}_\mathbb{X}$ is the excess of (spatial) regularity of the latter bound over the critical threshold, see \ref{['eq:excess_criticality_intro']}.
• Figure 3: Visualization of the behaviour of $u$ around the regular time $t_0$, where $\varepsilon\in (0,1)$. The red box $I_0^\varepsilon$ represents the region in the time-sample space where $u$ is regular for the $\mathbb{X}$-setting.

Theorems & Definitions (42)

• Theorem 1.1: Bounds on singular times for 3D stochastic NSEs -- Informal version of Theorems \ref{['t:singular_times_SNS']} and \ref{['t:singular_times_SNS_2']}
• Theorem 1.2: Bounds on singular times: Abstract formulation -- Informal version of Theorem \ref{['t:singular_times_SPDEs']}
• Lemma 2.1
• Remark 2.2: Lack of $\sigma$-subadditivity of the Minkowski content
• Lemma 2.3
• proof
• Definition 2.7: Local, unique and maximal in the $\mathbb{X}$-setting
• Theorem 2.8: Local well-posedness in the $\mathbb{X}$-setting
• Definition 3.2: Regular and singular times
• Definition 3.3: $\varepsilon$-regular and $\varepsilon$-singular times