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This paper discusses the solvability (global in time) of the initial-boundary value problem of the Navier–Stokes equations in the half space when the initial data \( h\in {\dot{B}}_{q \sigma }^{\alpha -\frac{2}{q}}({{\mathbb {R}}}^n_+)\) and the boundary data \( g\in \dot{ B}_q^{\alpha -\frac{1}{q},\frac{\alpha }{2}-\frac{1}{2q}}({{\mathbb {R}}}^{n-1}\times {\mathbb R}_+) \) with \(g_n\in \dot{B}^{\frac{1}{2} \alpha }_q ({{\mathbb {R}}}_+; \dot{B}^{-\frac{1}{q}}_q ({{\mathbb {R}}}^{n-1}))\cap L^q({\mathbb R}_+;{\dot{B}}^{\alpha -\frac{1}{q}}({{\mathbb {R}}}^{n-1}))\), for any \(0<\alpha <2\) and \(q =\frac{n+2}{\alpha +1}\). Compatibility condition (1.3) is required for h and g.

Initial-boundary value problem of the Navier–Stokes equations in the half space with nonhomogeneous data