TY - JOUR

T1 - Stability of stationary solutions to the three-dimensional Navier-Stokes equations with surface tension

AU - Watanabe, Keiichi

N1 - Funding Information:
Funding information : This research was partly supported by JSPS KAKENHI Grant Numbers 20K22311 and 21K13826 and the Waseda University Grant for Special Research Projects (Project number: 2021C-583).
Publisher Copyright:
© 2023 Keiichi Watanabe.

PY - 2023/1/1

Y1 - 2023/1/1

N2 - 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.

AB - 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.

KW - free boundary problems

KW - maximal regularity

KW - Navier-Stokes equations

KW - stability

KW - surface tension

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U2 - 10.1515/anona-2022-0279

DO - 10.1515/anona-2022-0279

M3 - Article

AN - SCOPUS:85145608327

VL - 12

JO - Advances in Nonlinear Analysis

JF - Advances in Nonlinear Analysis

SN - 2191-9496

IS - 1

M1 - 20220279

ER -