Global solutions for small data to the Hele-Shaw problem

P. Constantin, M. Pugh

Research output: Contribution to journalArticle

86 Scopus citations

Abstract

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 non-local 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.

Original languageEnglish (US)
Pages (from-to)393-415
Number of pages23
JournalNonlinearity
Volume6
Issue number3
DOIs
StatePublished - May 1993
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

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