On the Navier-Stokes equations on surfaces

TL;DR

This paper studies the Navier-Stokes equations on smooth surfaces, establishing well-posedness, characterizing equilibria as Killing fields, and proving stability and exponential convergence of solutions.

Contribution

It introduces a framework for analyzing Navier-Stokes flows on surfaces, characterizes equilibria, and proves stability and convergence results.

Findings

Well-posedness in $L_p$-$L_q$ framework

Equilibria characterized as Killing vector fields

Solutions near equilibrium are globally stable and exponentially converge

Abstract

We consider the motion of an incompressible viscous fluid that completely covers a smooth, compact and embedded hypersurface $\Sigma$ without boundary and flows along $\Sigma$. Local-in-time well-posedness is established in the framework of $L_p$-$L_q$-maximal regularity. We characterize the set of equilibria as the set of all Killing vector fields on $\Sigma$ and we show that each equilibrium on $\Sigma$ is stable. Moreover, it is shown that any solution starting close to an equilibrium exists globally and converges at an exponential rate to a (possibly different) equilibrium as time tends to infinity.