Dynamics of a complex interface

We study the motion of the interface between two fluids in a pressure field. A more viscous fluid surrounds a finite region filled with a less viscous fluid whose pressure is constant. The more viscous fluid is incompressible and moves with a velocity proportional to the gradient of its pressure. The pressure jump across the interface between the fluids is proportional to the curvature of the interface. The proportionality constant is the surface tension. In the two-dimensional case the interface can be described by a complex function which is analytic in the exterior of the unit circle. The dynamics are governed by a nonlocal nonlinear equation for this function. If all the singularities of the function are situated near the origin then the nonlocal interactions contribute little to the evolution of the singularities. In this case the nonlocal evolution equation can be approximated by a local nonlinear one. We prove that this equation has solutions with certain uniform behavior as the surface tension is allowed to become vanishingly small.