Almost sure existence of global weak solutions for incompressible generalized Navier-Stokes equations
Yuan-Xin Lin, Ya-Guang Wang
TL;DR
The paper addresses the global weak solvability of the incompressible generalized Navier–Stokes equations with fractional diffusion on the torus for rough initial data. It adopts a probabilistic approach by randomizing the initial data to obtain almost sure control of the linear evolution h^ω, then decomposes the solution as u = h^ω + w and solves for the nonlinear remainder w via a fixed-point argument on a short time interval, followed by an energy-based extension to all times. Under α in (2/3,1] and initial data u0 in dotH^s with s in (1-2α,0), the authors prove the almost sure global existence of weak solutions for a full-measure set of ω, and establish a weak–strong uniqueness mechanism to ensure well-posedness in the constructed framework. This work extends prior results at α = 1 to the generalized fractional-diffusion regime, highlighting the role of randomization in achieving global solvability in super-critical settings and contributing a probabilistic well-posedness paradigm for fractional Navier–Stokes equations.
Abstract
In this paper we consider the initial value problem of the incompressible generalized Navier-Stokes equations in torus $\mathbb{T}^d$ with $d \geq 2$. The generalized Navier-Stokes equations is obtained by replacing the standard Laplacian in the classical Navier-Stokes equations by the fractional order Laplacian $-(-Δ)^\al$ with $\al \in \left( \frac{2}{3},1 \right]$. After an appropriate randomization on the initial data, we obtain the almost sure existence of global weak solutions for initial data being in $\Dot{H}^s(\mathbb{T}^d)$ with $s\in (1-2\al,0)$.