On the two-phase Navier–Stokes equations with surface tension

Abstract

The two-phase free boundary problem for the Navier-Stokes system is considered in a situation where the initial interface is close to a halfplane. By means of L p -maximal regularity of the underlying linear problem we show local well-posedness of the problem, and prove that the solution, in particular the interface, becomes instantaneously real analytic.

Report on a local in time solvability of free surface problems for the Navier–Stokes equations with surface tension

“…In fact, all the results mentioned above, except for Moglilevskiȋ and Solonnikov [5] and Prüss and Simonett [9], are obtained in some Sobolev-Slobodetskii space W…”

Thin-film approximations of the two-phase Stokes problem

“…Moreover, the steady-state are exponentially stable: strong solutions converge at an exponential rate towards a flat steady-state, which is uniquely determined by the volume of the two fluids, if they are initially close to it. Finally, we stress that though the existence of analytic solutions for the one-phase Stokes problem [16][17][18][19] and for the two-phase Navier-Stokes problem with one free interface [20][21][22] is well-established, the well-posedness of the two-phase Stokes problem with two free interfaces (the system (2.1)) considered herein has not been studied analytically yet. Therefore, the question of convergence of solutions of the latter problem towards solutions of its thin film approximation is left as an open problem.…”

The Spin-Coating Process: Analysis of the Free Boundary Value Problem

“…The two-phase problem without rotational effects was investigated by Denisova in [9,10], by Tanaka in [41] and by Prüss and Simonett in [28,29].…”

“…On the top part, our problem differs from known one-or two-phase flow models through Coriolis and centrifugal force. Well-posedness results in the non-rotating setting for one-phase flows with surface tension are due to Solonnikov [35][36][37][38][39][40] and Shibata and Shimizu [31][32][33] and Prüss and Simonett [29]. In the setting of spin-coating it is natural to consider infinite layer-like domains.…”