Existence of bifurcating quasipatterns in steady Bénard-Rayleigh convection
The authors prove the existence of bifurcating quasipatterns in the steady Bénard-Rayleigh convection problem. The proof adapts a similar method, which was used to establish quasipatterns for the Swift-Hohenberg equation. The solution is approximated by a truncated power series expansion, which was formally obtained in a previous paper, but which diverges in general. After short sections introducing the physical problem, the quasilattices and the corresponding function spaces and operators, the properties of the convection problem for different boundary conditions are worked out in detail. Next the critical parameter value of the system and the corresponding wave vector are obtained and the initial series expansion for the solution is carried out. At fourth order of the series expansion the small divisor problem occurs and the authors have to use sophisticated methods to obtain valid solutions. They have to overcome the loss of self-adjointness of the linearization and to obtain a set of ``good parameter'' values, for which the ``range equation'' is solvable. Finally they obtain the solution by solving the bifurcation equation for ``good parameter'' values.
scientific article; zbMATH DE number 7006095
scientific article