Almost sure existence of global weak solutions for incompressible generalized Navier-Stokes equations

Yuan-Xin Lin; Ya-Guang Wang

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arXiv · 2502.15273

Almost sure existence of global weak solutions for incompressible generalized Navier-Stokes equations

Yuan-Xin Lin·Ya-Guang Wang

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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 $-(-\Delta)^\al$ 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 $s\in (1-2\al,0)$.

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Yuan-Xin Lin, Ya-Guang Wang. 2025-02-21. Almost sure existence of global weak solutions for incompressible generalized Navier-Stokes equations. https://arxiv.org/abs/2502.15273

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