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 ∈ L(R+ × (0, T )) with ∇u ∈ L q 2 loc (R+ × (0, T )) for a short time interval when the initial data h ∈ B − 2 q q (R+) and the boundary data g ∈ L(0, T ;B − 1 q q (R)) + L(R;B − 1 2q q (0, T )) with normal component gn ∈ L (0, T ; Ḃ − 1 q q (R)), n+ 2 < q < ∞ are given. 2000 Mathematics Subject Classification: primary 35K61, secondary 76D07.