Hölder Space Theory for the Rotation Problem of a Two-Phase Drop

Abstract

We investigate a uniformly rotating finite mass consisting of two immiscible, viscous, incompressible self-gravitating fluids which is governed by an interface problem for the Navier–Stokes system with mass forces and the gradient of the Newton potential on the right-hand sides. The interface between the liquids is assumed to be closed. Surface tension acts on the interface and on the exterior free boundary. A study of this problem is performed in the Hölder spaces of functions. The global unique solvability of the problem is obtained under the smallness of the initial data, external forces and rotation speed, and the proximity of the given initial surfaces to some axisymmetric equilibrium figures. It is proved that if the second variation of the energy functional is positive and mass forces decrease exponentially, then small perturbations of the axisymmetric figures of equilibrium tend exponentially to zero as the time t→∞, and the motion of liquid mass passes into the rotation of the two-phase drop as a solid body.

Keywords

• two-phase liquid problem with mass forces
• stability of a solution
• viscous incompressible self-gravitating fluids
• interface problem for the Navier–Stokes system
• Hölder spaces
• exponential decay