Global existence and decay for solutions of the Hele-Shaw flow with injection

C. H Arthur Cheng*, Daniel Coutand, Steve Shkoller

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

9 Citations (Scopus)


We examine the stability and decay of the free boundary perturbations in a Hele-Shaw cell under the injection of fluid. In particular, we study the perturbations of spherical boundaries as time t → +∞. In the presence of positive surface tension, we examine both slow and fast injection rates.When fluid is injected slowly, the perturbations decay back to an expanding sphere exponentially fast, while for fast injection, the perturbation decays to an expanding sphere with an algebraic rate. In the absence of surface tension, we study the case of a constant injection rate, and prove that perturbations of the sphere decay like (1 + t)-1/2+∈ for ∈ > 0 small.

Original languageEnglish
Pages (from-to)297-338
Number of pages42
JournalInterfaces and Free Boundaries
Issue number3
Publication statusPublished - 1 Jan 2014


  • Hele-Shaw
  • Hele-Shaw free boundary problems
  • Interface problems
  • Stability
  • Surface tension


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