Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 38, 2013, 259-286
KTH Royal Institute of Technology, Department of Mathematics
SE-100 44, Stockholm, Sweden; gbjorn 'at' kth.se
KTH Royal Institute of Technology, Department of Mathematics
SE-100 44, Stockholm, Sweden; ylli 'at' kth.se
Abstract. We study the dynamics of roots of f'(\zeta,t), where f(\zeta,t) is a locally univalent polynomial solution of the Polubarinova-Galin equation for the evolution of the conformal map onto a Hele-Shaw blob subject to injection at one point. We give examples of the sometimes complicated motion of roots, but show also that the asymptotic behavior is simple. More generally we allow f'(\zeta,t) to be a rational function and give sharp estimates for the motion of poles and for the decay of the Taylor coefficients. We also prove that any global in time locally univalent solution actually has to be univalent.
2010 Mathematics Subject Classification: Primary 30C15, 34M30, 37N10, 76D27.
Key words: Hele-Shaw flow, Laplacian growth, Polubarinova-Galin equation, Löwner-Kufarev equation, root dynamics, pole dynamics.
Reference to this article: B. Gustafsson and Y.-L. Lin: On the dynamics of roots and poles for solutions of the Polubarinova-Galin equation. Ann. Acad. Sci. Fenn. Math. 38 (2013), 259-286.
doi:10.5186/aasfm.2013.3802
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