Constantin, P.; Pugh, M. Global solutions for small data to the Hele-Shaw problem. (English) Zbl 0808.35104 Nonlinearity 6, No. 3, 393-415 (1993). 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. Cited in 58 Documents MSC: 35Q35 PDEs in connection with fluid mechanics 35K55 Nonlinear parabolic equations 35R35 Free boundary problems for PDEs 76D99 Incompressible viscous fluids Keywords:motion of an interface between two fluids; nonlinear asymptotic stability; well-posedness of the Cauchy problem PDFBibTeX XMLCite \textit{P. Constantin} and \textit{M. Pugh}, Nonlinearity 6, No. 3, 393--415 (1993; Zbl 0808.35104) Full Text: DOI Link