Time-periodic Poiseuille-type solution with minimally regular flow rate - Kaulakytė, Kristina; Kozulinas, Nikolajus; Pileckas, Konstantin (Nonlinear Analysis: Modelling and Control, 2021)
The nonstationary Navier–Stokes equations are studied in the infinite cylinder Π = x = (x’, xn) ∈ Rn: x’ ∈ σ ∈ R n – 1: – ∞ < xn < ∞, n = 2, 3 under the additional condition of the prescribed time-periodic flow-rate (flux) F(t). It is assumed that the flow-rate F belongs to the space L2(0, 2π), only. The time-periodic 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 time-periodic heat equation with a specific over-determination condition. The existence and uniqueness of a solution to this problem is proved.