On the large-distance asymptotics of steady state solutions of the Navier–Stokes equations in 3D exterior domains

Abstract

We identify the leading term describing the behavior at large distances of the steady state solutions of the Navier-Stokes equations in 3D exterior domains with vanishing velocity at the spatial infinity.

Generalized Scale-Invariant Solutions to the Two-Dimensional Stationary Navier--Stokes Equations

“…In three dimensions, Nazarov & Pileckas (1999, 2000 proved that the asymptotic behavior of solutions of (8) is a scale-invariant solution. Then Korolev & Šverák (2011) simplified the proof by showing directly that, in this case, the Landau solution is the correct asymptotic behavior of any solution bounded by (1 + |x|)…”

Time-Periodic Solutions to the Navier-Stokes Equations

“…In the case λ = 0, an asymptotic expansion was available only much later, and only for solutions corresponding to "small" data. This result is due to Korolev andŠverak, who showed in [35] that a Leray solution to (1.16) satisfies 18) where Γ α Landau is a so-called Landau solution, and R(x) again a remainder that decays faster as |x| → ∞ than Γ α Landau (x). A closed-form expression can be given for Γ α Landau .…”

“…If we further assume that f is sufficiently small, we can use a recent result by Korolev anď Sverák [35] to derive u / ∈ L ∞ 0, T ; L 2 (R 3 ) 3 also in the case λ = 0.…”

On Stability of Steady Circular Flows in a Two-Dimensional Exterior Disk

“…In the case of three-dimensional (fixed) exterior domains the unique existence of stationary Navier-Stokes flows satisfying the decay estimate O(|x| −1 ) (for velocity) as |x| → ∞ is proved by Finn [12], Galdi-Simader [16], Novotny and Padula [39], and Borchers and Miyakawa [3] under some smallness and decay conditions on given data, and in Korolev andŠverák [29] the asymptotic profile at spatial infinity is shown to be the Landau solution. The existence theory in the Lorenz spaces has been established by Kozono and Yamazaki [33].…”