Existence of local suitable weak solutions to the Navier-Stokes equations for initial data in $L^{2}_{\rm loc} (\mathbb{R}^3)$

TL;DR

This paper proves the local existence of suitable weak solutions to the Navier-Stokes equations with initial data in a local L^2 space, including conditions for global existence and partial regularity results.

Contribution

It establishes the existence of local weak solutions for initial data in L^2_{loc} and demonstrates global existence under specific conditions, along with partial regularity results.

Findings

Existence of local suitable weak solutions for initial data in L^2_{loc}

Global weak solutions when a certain parameter C=0

Partial regularity in the sense of Caffarelli-Kohn-Nirenberg

Abstract

We consider the Navier-Stokes equations in $\mathbb{R}^3$ subject to the initial condition with initial velocity field in $L^{2}_{\rm loc} (\mathbb{R}^3)$ such that $\limsup_{R \to +\infty } R^{-1} \|u_{0} \|_{ L^{2}(B(R))} < +\infty$. Our aim is to show the local existence of a weak solution, global existence of weak solution if $C=0$ and the partial regularity in the sense of Caffarelli-Kohn-Nirenberg.