On the Navier-Stokes equations on surfaces

2020/05/02 by Jan Pruess, Pruess, Jan, Gieri Simonett +3 · 2 citations
Engineering · Mathematics · #35B40 #35Q30 #35Q35 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2005.00830

openalex publication_date 2020/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the motion of an incompressible viscous fluid that completely covers a smooth, compact and embedded hypersurface Σ without boundary and flows along Σ. 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 Σ and we show that each equilibrium on Σ 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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• Self-avoiding fluid deformable surfaces
• Difference estimate for weak solutions to the Navier-Stokes equations in a thin spherical shell and on the unit sphere

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