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Lemniscates do not survive Laplace growth. (English) Zbl 1236.30006
The Polubarinova-Galin equation models the movement of the boundary between two fluids in the plane when a sink or source is present. (For background material see [B. Gustafsson and A. Vasil’ev, Conformal and potential analysis in Hele-Shaw cells. Advances in Mathematical Fluid Mechanics. Basel: Birkhäuser (2006; Zbl 1122.76002)].) In the present paper it is shown that the only polynomial lemniscates moving under this process of Laplacian growth (in fact, under a somewhat larger class of processes which are invariant under time reversal) are concentric circles.

30C20 Conformal mappings of special domains
35R37 Moving boundary problems for PDEs
76D27 Other free boundary flows; Hele-Shaw flows
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