Niveau: Supérieur, Doctorat, Bac+8
Quasipatterns in steady Benard-Rayleigh convection Gerard Iooss I.U.F., Universite de Nice, Labo J.A.Dieudonne Parc Valrose, F-06108 Nice, France August 4, 2009 Abstract Quasipatterns in the steady Benard-Rayleigh convection problem are considered. These are two-dimensional patterns, quasiperiodic in any horizontal direction, invariant under horizontal rotations of angle 2π/Q. As with problems involving quasiperiodicity, there is a small divisor problem. In this paper, we consider all cases with an even number Q ≥ 8. We prove that a formal solution, given by a divergent series, may be used to build a smooth quasiperiodic convection solu- tion which is an approximate solution of the Benard-Rayleigh system, up to an exponentially small error. Keywords: Rayleigh-Benard convection, bifurcations, quasipattern, small divisors, Gevrey series AMS: 35B32, 35C20, 40G10, 52C23 1 Introduction Quasipatterns are two-dimensional patterns which have no translation sym- metries and are quasiperiodic in any spatial direction (see figure 1). The spatial Fourier transforms of quasipatterns have discrete rotational order (most often, 8, 10 or 12-fold) and were first discovered in nonlinear pattern- forming systems in the Faraday wave experiment [3, 5], in which a layer of fluid is subjected to vertical oscillation.
- convection
- momentum equation
- small error
- only considering steady
- lyapunov-schmidt reduction
- steady benard-rayleigh
- dimensional pattern