Initial and boundary values for [formula omitted] solution of the Navier–Stokes equations in the half-space.

In this paper, we study the initial and boundary value problem of the Navier–Stokes equations in the half-space. We prove the existence of weak solution u ∈ L α q ( 0 , ∞ ; L p ( R + n ) ) , α = 1 2 ( 1 − n p − 2 q ) ≥ 0 , n < p < ∞ with ∇ u ∈ L loc q 2 ( 0 , ∞ ; L loc p 2 ( R + n ) ) for the solenoidal initial data h ∈ B ˙ p q − 1 + n p ( R + n ) and the boundary data g ∈ L α q ( 0 , ∞ ; B ˙ p p − 1 p ( R n − 1 ) ) when ‖ h ‖ B ˙ p q − 1 + n p ( R + n ) + ‖ g ‖ L α q ( 0 , ∞ ; B ˙ p p − 1 p ( R n − 1 ) ) is small enough. Moreover, the solution is unique in the class L α q ( 0 , T ; L p ( R + n ) ) for any T ≤ ∞ if α > 0 and for some T < ∞ if α = 0 . [ABSTRACT FROM AUTHOR]