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

TL;DR

This work connects the incompressible NSE with fractional dissipation to a probabilistic DSY cascade, revealing a self-similar stochastic representation that governs the mean-field solution in the supercritical regime $\gamma\in(\tfrac{1}{2},\tfrac{d+2}{4})$. By analyzing explosion, non-explosion, and hyper-explosion of the DSY cascade through detailed ratio-$R$ statistics and angular geometry, the authors delineate parametric phase transitions and establish a majorizing principle linking the stochastic recursion to a Montgomery–Smith-type equation. In 2D, the bilinear structure admits a closed-form minimal solution and symmetry-based averaging, enabling continuation beyond a finite critical time for certain initial data and yielding global behavior for vortex-like flows. Overall, the paper provides a probabilistic framework for NSE with fractional dissipation, clarifying how stochastic explosion scenarios correspond to (non)uniqueness and blow-up phenomena, and showing how symmetry can salvage global solutions in low dimensions.

Abstract

We identify a probabilistic structure known as self-similar doubly stochastic Yule (DSY) cascade associated with the deterministic Navier-Stokes equations (NSE) in $\mathbb{R}^d$ with fractional dissipation $(-Δ)^γ$. Interestingly, such a structure is well-defined only in the scaling-supercritical regime $γ\in(\frac{1}{2},\frac{d+2}{4})$. We then characterize parametric regions of $(d,γ)$ that correspond to the non-explosive, explosive, and hyperexplosive behavior 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 thanks to 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. Notably, 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 employing a notion of averaging.

Figures (8)

• Figure 1: Two halves of the plane
• Figure 2: Cascade figure that illustrates $\{{Y}_{v}\}_{v\in\mathbb{T}}$ and $\{W_{v}\}_{v\in\mathbb{T}}$.
• Figure 3: Cascade that is nonexplosive by time $t$.
• Figure 4: Kelvin inversion $\eta\mapsto\eta^*$, where $W_1=\eta$ and $\eta^*=\frac{1}{|\eta|^2}\eta$.
• Figure 6: $d\gamma$-diagram. The horizontal $d$-scale is discretized with stepsize 0.1, and the vertical $\gamma$-scale is discretized with stepsize 0.01. Two different ranges of $d$ are shown: $d\in[1.5,55]$ in the upper figure, and $d\in[1.7,12.2]$ in the lower figure.

Theorems & Definitions (64)

• Definition 1: Solution Process
• Definition 2
• Proposition 1
• proof
• Corollary 1
• proof
• Remark 1
• Definition 3
• Remark 2
• Proposition 2