Initial and boundary values for L-alpha(q)(L-p) solution of the Navier-Stokes equations in the half-space

摘要

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 is an element of L-alpha(q)(0, infinity; L-p(R-+(n)), alpha = 1/2(1-n/p-2/q) >= 0, n < p < infinity with del u is an element of L-loc(q/2)(0, infinity; L-loc(p/2)(R-+(n))) for the solenoidal initial data h is an element of(B)over dot(pq)(-1+n/p) (R-+(n)) and the boundary data g is an element of L-alpha(q)(0, infinity; (B)over dot(pp)(-1/p) (Rn-1)) when parallel to h parallel to((B)over dotpq-1/p (R+n)) + parallel to g parallel to(B)L-alpha((0, infinity:over dotpp-1/p (Rn-1))q is small enough. Moreover, the solution is unique in the class L-alpha(q)(0, T; L-p(R-+(n))) T; LP(118V) for any T <= infinity if alpha > 0 and for some T < infinity if alpha = 0. (C) 2016 Elsevier Inc. All rights reserved.