GLOBAL SOLVABILITY OF A PROBLEM GOVERNING THE MOTION

This is a common work with V. A. Solonnikov (St. Petersburg Department of Steklov Math. Institute, Russian Academy of Sciences).

Streszczenie:

We deal with the motion of two immiscible incompressible uids in a container. The liquids are separated by a close unknown interface on which surface tension is taken into account. We prove that this problem is uniquely solvable in an in nite time interval provided that the initial velocity of the liquids and mass forces are small, while the initial con guration of the inner uid is close to a ball. Moreover, we show that the velocity decays exponentially at in nity with respect to time and that the interface between the uids tends to a sphere of the certain radius. The proof is based on an exponential estimate of a generalized energy and on a local existence theorem for the problem.