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

Abstract

We consider the Navier-Stokes equations in ${\mathbb{R}}^3$ subject to the initial condition with initial velocity field in $L_{\mathrm{loc}}^2({\mathbb{R}}^3)$ such that ${\mathrm{limsup}}_{R\to +\infty }{R}^{-1}\Vert u_0{\Vert }_{L^2(B(R))}<+\infty$. 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.

Similar result is already obtained by Z. Bradshaw, I. Kukavica and T-P. Tsai (Indiana Univ. Math. J., vol. 7(2022), pp. 191-212