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.

我们证明了在“无刚性”边界条件下Bénard-Rayleigh对流的畴壁的存在。在最近的工作中（Haragus和Iooss在Arch Ration Mech Anal 239：733–781，2021中进行了研究），我们研究了在“刚性-刚性”和“自由-自由”边界条件下的分叉问题。在这三种情况下，为了证明存在性，我们使用空间动力学方法，其中将控制方程写为无限维动力学系统。中心流形定理表明，分叉的畴壁位于12维中心流形上，并且可以构造为将动力系统约束的周期解连接到该中心流形的异斜解。这些异诊所连接的存在性证明依赖于正常形式的分析，前导异质连接的构造以及隐函数定理。此处考虑的“无刚性”边界条件的情况与Haragus和Iooss（2021）中的两种情况之间的主要区别是控制方程式的垂直反射对称性丧失。这种对称性在Haragus和Iooss（2021）中得到了利用，它表明分叉的畴壁位于中心歧管的8维不变子流形上。因此，发现异质连接是8维而不是12维动力系统的解决方案。是控制方程式的垂直反射对称性的损失。这种对称性在Haragus和Iooss（2021）中得到了利用，它表明分叉的畴壁位于中心歧管的8维不变子流形上。因此，发现异质连接是8维而不是12维动力系统的解决方案。是控制方程式的垂直反射对称性的损失。这种对称性在Haragus和Iooss（2021）中得到了利用，它表明分叉的畴壁位于中心歧管的8维不变子流形上。因此，发现异质连接是8维而不是12维动力系统的解决方案。