Existence and exponential stability in L^r‐spaces of stationary Navier–Stokes flows with prescribed flux in infinite cylindrical domains

Abstract

SUMMARYWe prove existence, uniqueness and exponential stability of stationary Navier-Stokes flows with prescribed flux in an unbounded cylinder of R n , n 3, with several exits to infinity provided the total flux and external force are sufficiently small. The proofs are based on analytic semigroup theory, perturbation theory and L r − L q -estimates of a perturbation of the Stokes operator in L q -spaces.

A remark on Leray’s problem on stationary Navier–Stokes flows with large fluxes in infinite cylindrical domains

“…There is a number of papers dealing with stationary Leray's problem. Fundamental contribution to Leray's problem was made by Amick in [1], where the existence of unique weak solution to (1.2)-(1.4) was proved under a smallness assumption on the total flux m i=1 |Φi|, see also [2], [5], [6], [9], [10], and [13]- [17]. However, it has been shown, up to now, that Leray's problem is solved positively only under smallness assumptions on the total flux, and the problem for arbitrary large total flux is known as one of the most challenging problems in the theoretical fluid dynamics; for the Lerays and related problems we refer, in particular, to [7], Chap.…”

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

“…Proof First of all, we note that [17], Lemma 2.1, is valid for any domain with boundary of uniform C 1,1 -class.…”

“…In the stationary case a flux carrier was constructed in [7], Chapter 6, Section 1, for r = 2 and, applying the same idea, in [17] for q ∈ (1, ∞). In the instationary case a flux carrier can be constructed in a very similar way in principle, but we need a more refined argument.…”

Exponential stability for magneto-micropolar fluids

“…Subsequently, the arguments of [3] were adapted in [22] for the case of bounded domains (see also [32]). More recently, in [23] the authors studied the existence and exponential stability in L p of stationary Navier-Stokes flows with prescribed flux in infinite cylindrical domains through analytic semigroup theory, perturbation theory and L p -L q estimates for a perturbation of the Stokes operator in L q -spaces. For the micropolar fluid case, in [30] the authors used semigroup theory to obtain results on L p -stability.…”