Domain Walls for the Bénard–Rayleigh Convection Problem with “Rigid–Free” Boundary Conditions

Abstract

We prove the existence of domain walls for the Bénard–Rayleigh convection in the case of “rigid–free” boundary conditions. In the recent work (Haragus and Iooss in Arch Ration Mech Anal 239:733–781, 2021), we studied this bifurcation problem in the cases of “rigid–rigid” and “free–free” boundary conditions. In the three cases, for the existence proof we use a spatial dynamics approach in which the governing equations are written as an infinite-dimensional dynamical system. A center manifold theorem shows that bifurcating domain walls lie on a 12-dimensional center manifold, and can be constructed as heteroclinic solutions connecting periodic solutions of the restriction of the dynamical system to this center manifold. The existence proof for these heteroclinic connections then relies upon a normal form analysis, the construction of a leading order heteroclinic connection, and the implicit function theorem. The main difference between the case of “rigid–free” boundary conditions considered here and the two cases in Haragus and Iooss (2021), is the loss of a vertical reflection symmetry of the governing equations. This symmetry was exploited in Haragus and Iooss (2021) to show that bifurcating domain walls lie on an 8-dimensional invariant submanifold of the center manifold. Consequently, the heteroclinic connections were found as solutions of an 8-dimensional, instead of a 12-dimensional, dynamical system.