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

Abstract

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 R 3 {{\mathbb{R}}}^{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 L p {L}^{p} -in-time and L q {L}^{q} -in-space setting with ( p , q ) ∈ ( 2 , ∞ ) × ( 3 , ∞ ) \left(p,q)\in \left(2,\infty )\times \left(3,\infty ) satisfying 2 / p + 3 / q < 1 2\hspace{0.1em}\text{/}p+3\text{/}\hspace{0.1em}q\lt 1 , where the solution becomes real analytic, jointly in time and space. It is also proved that the solution converges exponentially to the equilibrium.