Kuznetsova, O. S. Polynomial solutions to the Hele–Shaw problem. (Russian, English) Zbl 0998.34071 Sib. Mat. Zh. 42, No. 5, 1084-1093 (2001); translation in Sib. Math. J. 42, No. 5, 907-915 (2001). The article is devoted to prove the solvability of the Hele-Shaw problem arising in free boundary problems of hydrodynamics produced by the injection of a fluid into a narrow channel.The author studies a class of polynomial solutions to the Hele-Shaw problem and, as a result, derives sufficient conditions on the numbers \(a_1,\dots,a_{n+1}\), \(a_i\in\mathbb{C}\), which guarantee that a strong polynomial solution \(w(z,t) = a_1(t)z + \dots + a_n(t)z^n\) with \(a_i(0) = a_i\) exists for all \(t\geq 0\). Moreover, certain qualitative properties of the solution obtained are presented. Reviewer: V.Grebenev (Novosibirsk) Cited in 3 Documents MSC: 34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain 34M25 Formal solutions and transform techniques for ordinary differential equations in the complex domain 34M55 PainlevĂ© and other special ordinary differential equations in the complex domain; classification, hierarchies 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 35Q51 Soliton equations Keywords:Hele-Shaw problem; polynomial solution; existence of a strong polynomial solution PDFBibTeX XMLCite \textit{O. S. Kuznetsova}, Sib. Mat. Zh. 42, No. 5, 1084--1093 (2001; Zbl 0998.34071); translation in Sib. Math. J. 42, No. 5, 907--915 (2001) Full Text: EuDML