Existence of Bifurcating Quasipatterns in Steady Bénard–Rayleigh Convection

Abstract

Extending the results obtained in the sixties for bifurcating periodic patterns, the existence of bifurcating quasipatterns in the steady Bénard–Rayleigh convection problem is proved. These are two-dimensional patterns, quasiperiodic in any horizontal direction, invariant under horizontal rotations of angle ${\pi/q}$. There is a small divisor problem for ${q \geqq 4}$.

Using the results of Berti–Bolle–Procesi in 2010, we adapt it to a Navier–Stokes system ruling the Bénard–Rayleigh convection problem. Our solution is approximated by the truncated power series which was formally obtained by Iooss in 2009, but which is divergent in general (Gevrey series). First, we formulate the problem in introducing a suitable parameter, able to move the spectrum of the linearized operator, as a whole, as for the Swift–Hohenberg PDE model. For using the Nash–Moser process, we are faced with the problem of inverting a linear operator which is the differential at a non zero point.

There are two new difficulties: (i) First, the extra dimension leading to a more complicated spectrum of the linear operator. This first difficulty leads to use specific projections for reducing the spectrum of the studied operator, which we want to invert, to a finite set very close to 0. (ii) The second difficulty is the fact that the linearization L^(N) at a non-zero point leads to a non-selfadjoint operator, contrary to what occurs in previous works. This is more serious, and leads to use the spectrum of L^(N)L^(N)* which depends mainly quadratically on the main parameter. A careful study of the “bad set”of parameters, with an assumption on the convexity of the eigenvalues of this operator, allows us to obtain a good estimate, as it is necessary for using the results of Berti et al. for solving ”the range equation”. We again use separation properties of the Fourier spectrum (see the Bourgain and Craig results) for obtaining an estimate in high Sobolev norms. It then remains to solve the one-dimensional “bifurcation equation.

For any ${q \geqq 4}$ , and provided that a weak transversality conjecture is realized, we prove the existence of a bifurcating convective quasipattern of order 2q, above the critical Rayleigh number.