Uniqueness of Steady Navier-Stokes Flows in Exterior Domains

Abstract

Abstract. We consider the uniqueness of stationary solutions to the Navier-Stokes equation in 3-dimensional exterior domains within the class u A L 3; y with 'u A L 3=2; y , where L 3; y and L 3=2; y are the Lorentz spaces. It is shown that if solutions u and v satisfy the conditions that u is small in L 3; y and v A L 3 þ L y , then u ¼ v. The proof relies upon the regularity theory for the perturbed Stokes equation.

Uniqueness of Solutions on the Whole Time Axis to the Navier-Stokes Equations in Unbounded Domains

“…Hence, we can prove Theorem 1 without assuming condition (1.3), provided that v ∈ Downloaded by [Georgetown University] at 00: 30 18 August 2015 S I and sup t<T u L 3 < min , instead of (1.5). From this observation, we notice that our uniqueness result improves that in [39].…”

“…Hence, the set of all functions having precompact range is much larger than the set of all almost periodic functions. In the present paper, we establish new uniqueness theorems for bounded continuous solutions having precompact range on the whole time axis, which improve our previous results in [14,15,38,39,45]. Moreover, we consider the uniqueness of solutions with (1.1) and solutions in weighted L spaces.…”

“…The second author [38,39] also proved similar uniqueness theorems for stationary solutions. In [39], he proved that if u and v are stationary solutions in L 3 with u v ∈ L 3/2 for the same force f , and if u is small in L 3 and v ∈ L 3 + L , then u = v.…”

A remark on the uniqueness of Kozono–Nakao’s mild $$L^3$$-solutions on the whole time axis to the Navier–Stokes equations in unbounded domains

“…the precompact range condition (PRC), then u = v. We note that similar uniqueness theorem was proven in [14,15,46] for time-periodic solutions, almost periodic-intime solutions and backward asymptotically almost periodic-in-time solutions, respectively. See also [39,40]. We first recall the uniqueness theorems given in [8] below.…”