A new proof of existence in the $L^3$-setting of solutions to the Navier–Stokes Cauchy problem

Abstract

We investigate the existence of solutions with initial datum $U_0$ in $L^3$. Our main goal is to establish the existence interval $(0,T)$, uniquely considering the size ${||U_0||}_3$ and the absolute continuity of ${|U_0(x)|}^3$.