Stability of stationary solutions to the three-dimensional Navier-Stokes equations with surface tension

Keiichi Watanabe*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

This article studies the stability of a stationary solution to the three-dimensional Navier-Stokes equations in a bounded domain, where surface tension effects are taken into account. More precisely, this article considers the stability of equilibrium figure of uniformly rotating viscous incompressible fluid in -3, which are rotationally symmetric about a certain axis. It is proved that this stability result can be obtained by the positivity of the second variation of the energy functional associated with the equation that determines an equilibrium figure, provided that initial data are close to an equilibrium state. The unique global solution is constructed in the Lp-in-time and Lq-in-space setting with (p, q) ϵ (2, ∞) × (3,∞) satisfying 2/p + 3/q < 1, where the solution becomes real analytic, jointly in time and space. It is also proved that the solution converges exponentially to the equilibrium.

Original languageEnglish
Article number20220279
JournalAdvances in Nonlinear Analysis
Volume12
Issue number1
DOIs
Publication statusPublished - 2023 Jan 1

Keywords

  • free boundary problems
  • maximal regularity
  • Navier-Stokes equations
  • stability
  • surface tension

ASJC Scopus subject areas

  • Analysis

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