Asymptotic Properties of Steady and Nonsteady Solutions to the 2D Navier-Stokes Equations with Finite Generalized Dirichlet Integral

Hideo Kozono, Yutaka Terasawa, Yuta Wakasugi

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)

Abstract

We consider the stationary and non-stationary Navier-Stokes equations in the whole plane R2 and in the exterior domain outside of the large circle. The solution v is handled in the class with ∇v ∈ Lq for q ≥ 2. Since we deal with the case q ≥ 2, our class is larger in the sense of spatial decay at infinity than that of the finite Dirichlet integral, that is, 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|−(1/q+1/q2)) and ∇v(x) = o(|x|−(1/q+1/q2)log|x|) as |x| → ∞, where ω ≡ rotv. As an application, we prove the Liouville-type theorem under the assumption that ω ∈ Lq(R2). For the non-stationary problem, a generalized Lq-energy identity is clarified. We also apply it to the uniqueness of the Cauchy problem and the Liouville-type theorem for ancient solutions under the assumption that ω ∈ Lq(R2 × I).

Original languageEnglish
Pages (from-to)1299-1316
Number of pages18
JournalIndiana University Mathematics Journal
Volume71
Issue number5
DOIs
Publication statusPublished - 2022

Keywords

  • Liouville-type theorems
  • Navier-Stokes equations

ASJC Scopus subject areas

  • General Mathematics

Fingerprint

Dive into the research topics of 'Asymptotic Properties of Steady and Nonsteady Solutions to the 2D Navier-Stokes Equations with Finite Generalized Dirichlet Integral'. Together they form a unique fingerprint.

Cite this