Existence of nonstationary Poiseuille type solutions under minimal regularity assumptions

Abstract

Existence and uniqueness of a solution to the nonstationary Navier–Stokes equations having a prescribed flow rate (flux) in the infinite cylinder Π={x=(x′,xn)∈Rn:x′∈σ⊂Rn−1,−∞<xn<∞,n=2,3} are proved. It is assumed that the flow rate F∈L2(0,T) and the initial data u0=(0,…,0,u0n)∈L2(σ). The nonstationary Poiseuille solution has the form u(x,t)=(0,…,0,U(x′,t)),p(x,t)=−q(t)xn+p0(t), where (U(x′,t),q(t)) is a solution of an inverse problem for the heat equation with a specific over-determination condition.