Incompressible Navier–Stokes flows with time-dependent prescribed fluxes in domains with cylindrical outlets to infinity

Abstract

We consider instationary Navier-Stokes equations with prescribed timedependent fluxes in a cylindrical domain ⊂ R n , n ≥ 3, with several exits to infinity. First, we prove existence and uniqueness of time-dependent Poiseuille flow in L qspaces on infinite straight cylinders over time interval (0, T ), 0 < T ≤ ∞, by obtaining an estimate of the pressure gradient using heat semigroups and techniques of Fourier multipliers. Then, based on sharp estimates in Besov spaces of the nonlinear term in the Navier-Stokes equations, we show the existence, uniqueness of a strong solution with prescribed time-dependent flux in each exit of in L p (0, T ; L q β ( )), p > 2, q ∈ ( n−1 2 , ∞), with an exponential weight along the axial directions of .

On a non‐stationary fluid flow problem in an infinite periodic pipe

“…The Poiseuille type solution (the velocity has a prescribed flux, and the pressure function is expressed by ) for the stationary Stokes and Navier–Stokes problems in domains with periodically varying section was found in , . The non‐stationary solutions with prescribed fluxes to Stokes and Navier–Stokes problems in infinite cylinders and in unbounded domains with cylindrical outlets to infinity were studied in , , , , . The existence of time periodic Poiseuille type solutions in the infinite cylinders was proved in , .…”