Let u be a weak solution of the Navier-Stokes equations in a smooth bounded domain Ω ⊆ ℝ³ and a time interval [0,T), 0 < T ≤ ∞, with initial value u₀, external force f = div F, and viscosity ν > 0. As is well known, global regularity of u for general u₀ and f is an unsolved problem unless we pose additional assumptions on u₀ or on the solution u itself such as Serrin’s condition ${\left|\right|u\left|\right|}_{L^s(0,T;L^q\left(\Omega \right))}<\infty$ where 2/s + 3/q = 1. In the present paper we prove several local and global regularity properties by using assumptions...

Nonlinear Schrödinger equations (NLS)${}_a$ with strongly singular potential ${a\left|x\right|}^{-2}$ on a bounded domain $\Omega$ are considered. If $\Omega ={ℝ}^N$ and $a>-{(N-2)}^2/4$, then the global existence of weak solutions is confirmed by applying the energy methods established by N. Okazawa, T. Suzuki, T. Yokota (2012). Here $a=-{(N-2)}^2/4$ is excluded because $D\left(P_{a\left(N\right)}^{1/2}\right)$ is not equal to $H^1\left({ℝ}^N\right)$, where $P_{a\left(N\right)}:=-\Delta -{(N-2)}^2/{\left(4\right|x|}^2)$ is nonnegative and selfadjoint in $L^2\left({ℝ}^N\right)$. On the other hand, if $\Omega$ is a smooth and bounded domain with $0\in \Omega$, the Hardy-Poincaré inequality is proved in J. L. Vazquez, E. Zuazua (2000)....