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Global solutions for small data to the Hele-Shaw problem. (English) Zbl 0808.35104

Summary: We analyse 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 existence for 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.

MSC:

35Q35 PDEs in connection with fluid mechanics
35K55 Nonlinear parabolic equations
35R35 Free boundary problems for PDEs
76D99 Incompressible viscous fluids
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