Geodesic
Global existence and decay for solutions of the Hele–Shaw flow with injection
C.H. Arthur Cheng1 ; Daniel Coutand2 ; Steve Shkoller3 orcid
1National Central University, Jhongli City (Taoyuan), Taiwan
2Heriot-Watt University, Edinburgh, UK
3University of Oxford, UK
Interfaces and free boundaries, Tome 16 (2014) no. 3, pp. 297-338

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DOI

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+ε for ε>0 small.
DOI : 10.4171/ifb/321
Classification : 35-XX, 76-XX
Mots-clés : Hele–Shaw, interface problems, stability, Hele–Shaw free boundary problems, surface tension

C.H. Arthur Cheng  1   ; Daniel Coutand  2   ; Steve Shkoller  3

1 National Central University, Jhongli City (Taoyuan), Taiwan
2 Heriot-Watt University, Edinburgh, UK
3 University of Oxford, UK 
C.H. Arthur Cheng; Daniel Coutand; Steve Shkoller. Global existence and decay for solutions of the Hele–Shaw flow with injection. Interfaces and free boundaries, Tome 16 (2014) no. 3, pp. 297-338. doi: 10.4171/ifb/321
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     title = {Global existence and decay for solutions of the {Hele{\textendash}Shaw} flow with injection},
     journal = {Interfaces and free boundaries},
     pages = {297--338},
     year = {2014},
     volume = {16},
     number = {3},
     doi = {10.4171/ifb/321},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/321/}
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