Stability for the 3D Navier–Stokes Equations with Nonzero far Field Velocity on Exterior Domains

Abstract

Concerning to the non-stationary Navier-Stokes flow with a nonzero constant velocity at infinity, just a few results have been obtained, while most of the results are for the flow with the zero velocity at infinity. The temporal stability of stationary solutions for the Navier-Stokes flow with a nonzero constant velocity at infinity has been studied by Enomoto and Shibata (J Math Fluid Mech 7:339-367, 2005), in L p spaces for p ≥ 3. In this article, we first extend their result to the case 3 2 < p by modifying the method in Bae and Jin (J Math Fluid Mech 10: [423][424][425][426][427][428][429][430][431][432][433] 2008) that was used to obtain weighted estimates for the Navier-Stokes flow with the zero velocity at infinity. Then, by using our generalized temporal estimates we obtain the weighted stability of stationary solutions for the Navier-Stokes flow with a nonzero velocity at infinity.Mathematics Subject Classification (2010). Primary 35Q30; Secondary 76D05.

The 3D Time-Dependent Oseen System: Link Between $$L^p$$-Integrability in Time and Pointwise Decay in Space of the Velocity

“…where Ω c := R 3 \Ω denotes an exterior domain, with Ω ⊂ R 3 being an open bounded set with Lipschitz boundary. This set is given, as are the quantities T 0 ∈ (0, ∞] and τ ∈ (0, ∞), as well as the function f : Ω c × (0, T 0 ) → R. The unknowns are the functions u : Ω c × (0, T 0 ) → R 3 and π : Ω c × (0, T 0 ) → R. The Oseen system is a linearization of the Navier-Stokes system with Oseen term…”

“…This is the reason why in [36], the restrictive integrability conditions mentioned above are imposed on ∇ x u|∂Ω × (0, T 0 ) and π|∂Ω × (0, T 0 ). In [10], [13] and [14], we circumvented this difficulty by solving an integral equation in a certain subspace of L 2 0, T 0 , L 2 (∂Ω) 3 . This approach provides a representation formula for solutions to (1.1) which does not contain the critical integrals mentioned above.…”

“…Maximal regularity cannot be expected to hold for solutions to the Oseen system (1.1). In fact, according to [20], the velocity part U of a solution (U, Π) to the Oseen resolvent system 3 and all λ ∈ C with λ > 0. As a consequence of this negative result, which arises since small values of |λ| are admitted, an analogous resolvent estimate cannot be expected to hold for solutions to the Oseen resolvent problem in Ω c , under whatever boundary conditions.…”

Navier–Stokes equations on exterior domains: review for the stability for the incompressible flow on exterior domains

“…Bae-Roh [5] extended Enomoto-Shibata's result for p < 3 and improved the estimates for p ≥ 3: Let 1 <r , r ≤ p < ∞ and u 0 ∈ L r ( ) ∩ L 3 ( ). They also assume that for a solution w of the system (0.3) there exist small positive numbers α 0 and β 0 such that…”

“…With the above results, Bae-Roh [5] derived the spatial-temporal convergence rates of nonstationary solutions to the stationary solutions: With the same assumptions in the above, we let 0 < σ < 1 2 be a number such that…”