Navier-Stokes Equations with Fractional Dissipation and Associated Doubly Stochastic Yule Cascades

Abstract

We introduce a self-similar doubly stochastic Yule (DSY) cascade associated with the deterministic Navier-Stokes equations (NSE) in $\mathbb{R}^d$ with fractional dissipation $(-\Delta)^\gamma$. Interestingly, such a structure is well-defined only in the scaling-supercritical regime $\gamma\in(\frac{1}{2},\frac{d+2}{4})$. We then characterize parametric regions of $(d,\gamma)$ that correspond to the stochastically explosive, non-explosive, hyperexplosive, non-hyperexplosive behaviors of the DSY cascade. Stochastic solution processes are constructed recursively, and their expectations yield solutions to the fractional NSE whenever these expectations exist. Explosion and geometric properties of the DSY cascade are then exploited to establish non-uniqueness and finite-time blowup results for a scalar partial differential equation associated with the fractional NSE using a majorization principle for stochastic solution processes. In the special case $d=2$, we derive a closed form for the solution process and prove the finite-time loss of integrability of the solution process for sufficiently large initial data. This lack of integrability does not necessarily imply finite-time blowup of solutions to the fractional NSE. Indeed, for vortex-flow initial data, we show that the solution can be continued beyond the time of integrability breakdown by averaging the stochastic solution processes in a way that creates symmetry cancellations.