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

Finite energy of generalized suitable weak solutions to the Navier-Stokes equations and Liouville-type theorems in two dimensional domains