On the Navier–Stokes equations on surfaces

Abstract

Abstract We consider the motion of an incompressible viscous fluid that completely covers a smooth, compact and embedded hypersurface $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Σ</mml:mi></mml:math> without boundary and flows along $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Σ</mml:mi></mml:math> . Local-in-time well-posedness is established in the framework of $$L_p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:math> - $$L_q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>L</mml:mi><mml:mi>q</mml:mi></mml:msub></mml:math> -maximal regularity. We characterize the set of equilibria as the set of all Killing vector fields on $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Σ</mml:mi></mml:math> , and we show that each equilibrium on $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Σ</mml:mi></mml:math> 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.