  Introducing a new notion of generalized suitable weak solutions, we first
prove validity of the energy inequality for such a class of weak solutions to
the Navier-Stokes equations in the whole space $\mathbb{R}^n$. Although we need
certain growth condition on the pressure, we may treat the class even with
infinite energy quantity except for the initial velocity. We next handle the
equation for vorticity in 2D unbounded domains. Under a certain condition on
the asymptotic behavior at infinity, we prove that the vorticity and its
gradient of solutions are both globally square integrable. As their
applications, Loiuville-type theorems are obtained.
