Elliptic and Parabolic Equations - Abstract

Prüß, Jan

Qualitative behavior of solutions for the two-phase Navier--Stokes equations with surface tension

The two-phase free boundary value problem for the isothermal Navier--Stokes equations is studied for general bounded geometries in absence of phase transitions, external forces and boundary contacts. It is shown that the problem is well-posed in an $L_p$-setting, and that it generates a local semiflow on the induced phase manifold. If the phases are connected, the set of equilibria of the system form an $(n+1)$-dimensional manifold, we prove that each equilibrium is stable. Furthermore, we show that each solution which does not develop singularites exists globally and converges to an equilibrium as time goes to infinity.