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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.

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 Painlevé 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
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