Asymptotic properties of steady and nonsteady solutions to the 2D Navier-Stokes equations with finite generalized Dirichlet integral

Abstract

We consider the stationary and non-stationary Navier-Stokes equations in the whole plane R 2 and in the exterior domain outside of the large circle. The solution v is handled in the class with ∇v ∈ L q for q ≥ 2. Since we deal with the case q ≥ 2, our class is larger in the sense of spacial decay at infinity than that of the finite Dirichlet integral, i.e., for q = 2 where a number of results such as asymptotic behavior of solutions have been observed.For the stationary problem we shall show that ω(x) = o(|x|2 ) as |x| → ∞, where ω ≡ rot v. For the non-stationary problem, a generalized L q -energy identity is clarified. As an application, we prove the Liouville type theorems under the assumption that ω ∈ L q (R 2 ) and ω ∈ L q (0, T ; L q (R 2 )) for the stationary and the non-stationary equations, respectively.

Classification of solutions of the 2D steady Navier-Stokes equations with separated variables in cone-like domains

“…which has characteristic equation (8). Case 2.1: If δ > 0, equation (8) has two different real roots m, n (m > n) and the general solution of ( 13) is expressed as follows…”

“…where λ ± µi (µ = 0) are the complex roots of equation (8). Since ϕ 1 and ϕ 2 are linearly independent, then C 2 1 + C 2 2 = 0. ϕ 1 , ϕ 2 ∈ C 0 at r = 0, then k = λ > 0.…”

“…When the smooth solution u is bounded, a Liouville theorem being more in the spirit of the classical one for entire analytic functions was obtained by in [7] as a byproduct of their work on the non-stationary case. If ∇u ∈ L q (R 2 ) with 1 < p < ∞, the constant u follows by the first author in [12] by using the growth estimate of D 1,q functions (see also [8] for another approach). As suggested by Fuchs-Zhong in [3]: "Suppose that lim |x|→∞ |x| −1 |u(x)| = 0.…”

Asymptotic properties of generalized D-solutions to the stationary axially symmetric Navier-Stokes equations

“…See also [1,14] for some related improvements. Recently Kozono-Terasawa-Wakasugi [18] showed that solutions of (1.1) in 2D space with (1.2) (2 < q < +∞) satisfy a priori estimates u(x) = o(|x| 1−2/q ) and w(x) = o(|x|…”