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arXiv · 2005.00830

On the Navier-Stokes equations on surfaces

Jan Pruess·Gieri Simonett·Mathias Wilke
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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.

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Jan Pruess, Gieri Simonett, Mathias Wilke. 2020-05-02. On the Navier-Stokes equations on surfaces. https://arxiv.org/abs/2005.00830

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