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Strongly \(\varPhi \)-like functions of order \(\alpha \) in two-dimensional free boundary problems. (English) Zbl 1379.76013

Summary: We apply certain results in the theory of univalent functions to investigate the time evolution of the free boundary of a viscous fluid for a planar flow problem in the Hele-Shaw cell model under injection. To this end, we prove that the property of strongly \(\varPhi \)-likeness of order \(\alpha \in (0, 1]\) is preserved in time for both inner and outer problems, under the assumption of nonzero small surface tension.

MSC:

76D27 Other free boundary flows; Hele-Shaw flows
76D45 Capillarity (surface tension) for incompressible viscous fluids
76M40 Complex variables methods applied to problems in fluid mechanics
35R35 Free boundary problems for PDEs
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