$L^2$-setting theory for the solutions to 2D Navier-Stokes equations: some new estimates

For an unbounded planar domain $\Omega$, we prove that the solution $u$ to the IBVP for the Navier-Stokes equations with initial datum $u_0\in L^2(\Omega)$ satisfies the following estimate \begin{equation*}
\int_0^{+\infty}\|u(\tau)\|_\infty^2d\tau\le c(1+\|u_0\|_2^2)\|u_0\|_2^2, \end{equation*} established by R. Farwig and Y. Giga [Algebra i Analiz, 36, 289-307 (2024)] for bounded domains. Moreover, we show that \begin{equation*}
\int_0^{+\infty}\tau\|P\Delta u(\tau)\|_2^2d\tau\le c(1+\|u_0\|_2^4 e^{c_1\|u_0\|_2^4})\|u_0\|_2^2, \end{equation*} and \begin{equation*}
\int_0^{+\infty}\tau\|u(\tau)\|_\infty^4d\tau\le c(1+\|u_0\|_2^4 e^{c_1\|u_0\|_2^4})\|u_0\|_2^4. \end{equation*}