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

Abstract

We study the stability of some exact stationary solutions to the two-dimensional Navier-Stokes equations in an exterior domain to the unit disk. These stationary solutions are known as a simple model of the flow around a rotating obstacle, while their stability has been open due to the difficulty arising from their spatial decay in a scale-critical order. In this paper we affirmatively settle this problem for small solutions. That is, we will show that if these exact solutions are small enough then they are asymptotically stable with respect to small L 2 perturbations.Keywords Navier-Stokes equations · two-dimensional exterior flows · scale-criticality · stability of stationary flows · flow around a rotating obstacle

Existence of the stationary Navier-Stokes flow in $\mathbb{R}^2$ around a radial flow

“…The largeness condition of µ plays a key role here, and in particular, it is not known in general whether or not the stationary solution exists when µ is not large. We note that the stability of µx ⊥ /|x| 2 for small perturbations was proved by M. [18] when |µ| is small enough; see also Higaki [10] for the genelarization of this stability result. Recently, Higaki-M.-Nakahara [12,11] showed the existence and uniqueness of stationary solutions around a rotating obstacle with small external forces and low speed of rotation, and later on, Gallagher-Higaki-M. [6] generalized it with high speed.…”

On stationary two‐dimensional flows around a fast rotating disk

“…The stability of these stationary solutions, though they are small in a scale-critical norm, is a difficult problem and is still largely open in the two dimensional case. The only known result is by Maekawa [20] for a specific case, which shows the local 2 stability of the explicit solution (1.3) in the original frame (1.1) when is small enough and Ω is the exterior to the unit disk as assumed in this paper. Few results are known in the case of nonsmall .…”

“…Then, applying the Hölder inequality we have 20) and the Young inequality for sequences implies that…”

The inviscid limit of Navier-Stokes equations for locally near boundary analytic data on an exterior circular domain

“…, where the functions K n (z) and I n (z) are modified Bessel functions with complex value z ∈ C (e.g., [17]), with W (I n , K n ) being the Wronskian determinant. The temporal Green function is then defined by taking the inverse Laplace transform in t of the kernel G ζ (r, r ′ ).…”

“…Remark 4.2. It is known from Section 2.2 of [17], that the Biot-Savart law 4.2 defines a unique velocity that decays at infinity, under the decaying assumption r 1−|n| ω n ∈ L 1 . In our current work, the vorticity satisfies the decaying assumption…”