Existence of nonstationary Poiseuille-type solutions under minimal regularity assumptions

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 ′ , x n ) ∈ R n : x ′ ∈ σ ⊂ R n - 1 , - ∞ < x n < ∞ , n = 2 , 3 } are proved. It is assumed that the flow rate F ∈ L 2 ( 0 , T ) and the initial data u 0 = ( 0 , … , 0 , u 0 n ) ∈ L 2 ( σ ) . The nonstationary Poiseuille solution has the form u ( x , t ) = ( 0 , … , 0 , U ( x ′ , t ) ) , p ( x , t ) = - q ( t ) x n + p 0 ( t ) , where ( U ( x ′ , t ) , q ( t ) ) is a solution of an inverse problem for the heat equation with a specific over-determination condition.