Global Solutions for Small Data to the Hele-Shaw Problem
with P. Constantin, Nonlinearity, 6(1993)393-416.
We analyze an equation governing the motion of an interface
between two fluids in a pressure field. In two dimensions,
the interface is described by a conformal
mapping which is analytic
in the exterior of the unit disc. This mapping obeys a nonlocal nonlinear
equation. When there is no pumping at infinity,
there is conservation of area and
contraction of the length of the interface. We prove global in time
small analytic perturbations
of the circle as well as nonlinear asymptotic stability of the steady circular
solution. The same method yields well-posedness of the Cauchy
problem in the presence of pumping.
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