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

Abstract: Parametric regions are identified in terms of the spatial dimension $d$ and the power $\gamma$ of the Laplacian that separate explosive from non-explosive regimes for the self-similar doubly stochastic Yule cascades (DSY) naturally associated with the deterministic fractional Navier-Stokes equations (FNSE) on $\mathbb{R}^d$ in the scaling-supercritical setting. Explosion and/or geometric properties of the DSY, are then used to establish non-uniqueness, local existence, and finite-time blow-up results for a scalar partial differential equation associated to the FNSE through a majorization principle at the level of stochastic solution processes. The solution processes themselves are constructed from the DSY and yield solutions to the FNSE upon taking expectations. In the special case $d=2$, a closed-form expression for the solution process of the FNSE is derived. This representation is employed to prove finite-time loss of integrability of the solution process for sufficiently large initial data. Notably, this lack of integrability does not necessarily imply blow-up of the FNSE solutions themselves. In fact, in the radially symmetric case, solutions can be continued beyond the integrability threshold by employing a modified notion of averaging.