Solvability of the initial–boundary value problem of the Navier–Stokes equations with rough data

In this paper, we study the initial and boundary value problem of the Navier–Stokes equations in the half space. We prove the unique existence of weak solution u∈Lq(R+n×(0,T)) with ∇u∈Llocq2(R+n×(0,T)) for a short time interval when the initial data h∈Bq−2q(R+n) and the boundary data g∈Lq(0,T;Bq−1q(Rn−1))+Lq(Rn−1;Bq−12q(0,T)) with normal component gn∈Lq(0,T;Ḃq−1q(Rn−1)), n+2