On square-in-time integrability of the maximum norm of a finite energy solution to the planar Navier–Stokes equations

The paper is devoted to the Navier–Stokes equations in a planar smooth bounded domain $\Omega$ under the no-slip boundary condition with initial velocity $u_0$ of finite kinetic energy, i.e., $u_0\in L^2$. The existence is proved for a unique weak solution $u\in L^2\left(0,T;L^\infty(\Omega)\right)$ satisfying the estimate
$$ \|u\|_{L^2\left(0,T;L^\infty(\Omega)\right)} \leq c \left(1+\|u_0\|_{L^2}\right) \|u_0\|_{L^2} $$
with some constant $c$ depending only on $\Omega$. Note that $H^1(\Omega)$ is not embedded into $L^\infty(\Omega)$ so that the argument is not based on the energy identity valid for $u$, but on a generalized Marcinkiewicz interpolation theorem. This estimate is extended to mild solutions of Serrin's class in $n$ dimensions provided that $u_0$ is in a solenoidal $L^n$ space.