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

By means of an $L^1-L^{\mathrm{\infty }}$ duality argument, it is proved that, in a suitable planar domain $\mathrm{\Omega }$, the solution $u$ to the IBVP associated to the Navier-Stokes equations, with initial datum $u_0\in L^2(\mathrm{\Omega })$, satisfies the following estimate $${({\int }_0^{+\mathrm{\infty }}\mathrm{\parallel }u(\tau ){\mathrm{\parallel }}_{\mathrm{\infty }}^2\text{d}\tau )}^{1\mathrm{/}2}\le c(1+\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_2)\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_2,$$ proved by R. Farwig and Y. Giga [Algebra i Analiz, 36, 289-307 (2024)] for bounded domains.