Remarks on the Strong Solvability of the Navier-Stokes Equations

Throughout this note m≥3 and either Ω=Rm, or Ω is a half-space of Rm, or Ω is a smooth domain in Rm with a compact boundary ∂Ω. We consider the following initial-boundary value problem (1) for the Navier-Stokes equations: ∇⋅v∂tv+(v⋅∇)v−νΔvvv(⋅,0)=0=−∇p=0=v0in Ω,in Ω,on ∂Ω,in Ω. Of course, there is no boundary condition if Ω=Rm. "In a recent paper [J. Math. Fluid Mech. 2 (2000), no. 1, 16--98] we investigated the strong solvability of (1) for initial data v0 belonging to certain spaces of distributions (modulo gradients). In this note we explain some of our main results in a very particular and simple setting. As usual, we concentrate on the velocity field v since the pressure field p is determined up to a constant by v.