Free Boundary Problems for the Navier-stokes Equations

A free boundary problem for the Navier-Stokes equations describes the flow of a viscous, incompressible fluid in a domain that is unknown or partially unknown. In this paper several results for flows in drops or in vessels are presented. The free boundary is governed by self-attraction or surface tension, and dynamic contact angles may occur. AMS-Classification: 76 D 05 , 35 R 35 § I. The Equations of Motion To determine the shape of a fluid body is a classical problem in mathematical physics. If the liquid rotates about a fixed axis and is moreover subject to self-attraction the problem was already investigated by I.Newton as a model for the figure of the earth. Since it was treated for the first time in the Philosophiae Naturalis Principia Mathematica 300 years ago it has stimulated research in various branches of mathematical analysis as for example potential theory, bifurcation theory for nonlinear integral equations, and more recently it was taken up again in connection with variational methods for free boundary problems, see e.g. Friedman IF2] Chap.4. According to Newton, s law the force of self-attraction equals DU(x) = D I ~ dy , where p = const in the density, D c ~3 the domain occupied by the fluid, and g the gravitational constant. If the body rotates about 3 the x -axis the centrifugal force is 2 r 2 DR(x) = D --~ (x) , where ~ denotes the angular velocity and r(x) = [(xl) 2 + (x2)2] I/2 the distance of a point x from the axis of rotation. With no other forces present the boundary ~ of D must be an equipotential surface: