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

TL;DR

This paper proves that solutions to the 2D Navier-Stokes initial boundary value problem in certain planar domains have a finite square-integrable maximum norm over time, extending previous results to unbounded domains.

Contribution

It establishes a square in time integrability estimate for the maximum norm of Navier-Stokes solutions in planar domains with finite energy initial data.

Findings

Solutions have finite time integral of maximum norm squared.

Extension of bounded domain results to certain unbounded domains.

Provides a duality-based proof technique.

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.