On the 2D initial boundary value problem for the Navier-Stokes equations: square in time integrability of the maximum norm of the solutions with finite energy

Abstract

By means of an $L^1-L^\infty$ duality argument, it is proved that, in a suitable planar domain $\Omega$, the solution $u$ to the IBVP associated to the Navier-Stokes equations, with initial datum $u_0\in L^2(\Omega)$, satisfies the following estimate $$ \left(\int_0^{+\infty}\|u(\tau)\|_\infty^2{\hbox {d}}\tau\right)^{1/2}\le c(1+\|u_0\|_2)\|u_0\|_2, $$ proved by R. Farwig and Y. Giga [Algebra i Analiz, 36, 289-307 (2024)] for bounded domains.