Initial and boundary values for Lαq(Lp) solution of the Navier–Stokes equations in the half-space

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∈Lαq(0,∞;Lp(R+n)), α=12(1−np−2q)≥0, n0α>0 and for some T<∞T<∞ if α=0α=0.