On stability of exterior stationary Navier-Stokes flows

About the Regularized Navier?Stokes Equations

“…Note that (2.3) has the form u = y + B(u, u), where the bilinear form is defined as 4) and y = S(t)u 0 + t 0 S(t − τ )IP F (τ ) dτ . Hence, using the classical Picard approach, which is based on Lemma 4.1 below, one can easily construct solutions in the space…”

Periodic solutions in unbounded domains for the Boussinesq system

“…Borchers and Miyakawa [2] established the following Helmholtz decomposition of the Lorentz spaces extending the operator P r to a bounded operator on L (r,d) (Ω), which we denote by P r,d . Setting…”

“…For simplicity, we shall abbreviate the projection operator and the Stokes and Laplace operators on Lorentz spaces by P , A, B, respectively. In view of [2], the operators…”

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].…”

“…The existence theory in the Lorenz spaces has been established by Kozono and Yamazaki [33]. The local L r stability of these stationary solutions is established in [3] and Kozono and Yamazaki [32], while the global L 2 stability is shown in [3]. The reader is referred to recent results by Karch, Pilarczyk, and Schonbek [28] and Hishida and Schonbek [24], where the global L 2 stability of small global solutions in L ∞ (0, ∞; L 3,∞ σ (R 3 )) is obtained.…”