Global in time solvability of the Navier-Stokes equations in the half-space

In this paper, we study the initial value problem of the Navier-Stokes equations in the half-space. Let. a solenoidal initial velocity be given in the function space (B) over dot(pq,0)(alpha-2/2)(R-+(n)) for 0 < alpha < 2, 1 < p, q < infinity with alpha + 1 = n/p + 2/q and 2/q < 1 + n/p.We prove the global in time existence of weak solution u is an element of L-q (0, infinity; (B) over dot(pq)(alpha)(R-+(n) )) boolean AND L-q0 (0, infinity; L-p0(R-+(n)))for some p < p0 < infinity and q < q0 < infinity with n/p0 + 2/q0 = 1, when the given initial velocity has small norm in function space (B) over dot(p0q0,0)(-2/q0)(R-+(n) ) (superset of (B) over dot(pq,0)(alpha-2/q)(R-+(n))) The solution is unique in the class L-q0 (0, infinity; L-P0 (R-+(n))). Pressure estimates are also given. (C) 2019 Elsevier Inc. All rights reserved.