We consider the nonstationary Navier-Stokes system in a smooth bounded domain Ω ⊂ R 3 with initial value u 0 ∈ L 2 σ (Ω). It is an important question to determine the optimal initial value condition in order to prove the existence of a unique local strong solution satisfying Serrin's condition. In this paper, we introduce a weighted Serrin condition that yields a necessary and sufficient initial value condition to guarantee the existence of local strong solutions u(·) contained in the weighted Serrin classMoreover, we prove a restricted weak-strong uniqueness theorem in this Serrin class.

This paper is concerned with a regularity criterion based on vorticity direction for Navier-Stokes equations in a three-dimensional bounded domain under the no-slip boundary condition. It asserts that if the vorticity direction is uniformly continuous in space uniformly in time, there is no type I blow-up. A similar result has been proved for a half space by Y. Maekawa and the first and the last authors (2014). The result of this paper is its natural but non-trivial extension based on L ∞ theory of the Stokes and the Navier-Stokes equations recently developed by K. Abe and the first author.

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