The last 100 days of the bieberbach conjecture

Classical mechanical systems with one-and-a-half degrees of freedom and Vlasov kinetic equation

“…whose compatibility conditions ∂ k (∂ i λ) = ∂ i (∂ k λ) lead to the Gibbons-Tsarev system (12). The celebrated Löwner equation initially appeared in 1923 as an ordinary nonlinear differential equation describing deformations of extremal univalent conformal mappings and was used in the solution of the famous Bieberbach Conjecture in 1984 (see an exposition of the history of this Conjecture in [10] and its relation to hydrodynamic reductions of Benney moment equations in [15]). Equations (12) and (13) were recently applied to the equations of Laplacian Growth, Dirichlet Boundary Problem and Hele-Shaw problem (see for instance [25]).…”

“…(the inverted asymptotic series (4)) into (16) yields Benney hydrodynamic chain (10) written in the conservative form…”

“…Thus, corresponding infinite set of equations (77) together with this condition V = a 0 implies Benney hydrodynamic chain (10). Now we study the pair (70)+(72).…”

“…In order to construct solutions of (14) we first need to prove the Egorov property of Benney hydrodynamic chain (10). This property is very important and many physical systems of hydrodynamic type integrable by the Generalized Hodograph Method possess this property (cf.…”

“…Expansion (4) is invariant under the scaling F → cF , p → cp and A k → c k+2 A k , where c is arbitrary constant. Thus, it is easy to see that Benney hydrodynamic chain (10) admits similarity reductions A k (x, t) = t −(β+1)(k+2) B k (z), where z = xt β . Substitution of this ansatz directly into Benney hydrodynamic chain (10) yields a chain of ordinary differential equations…”