Notes on symmetry in convective flows (Q2215514)

The Navier-Stokes equations in a 2D or 3D smooth domain are considered. Some symmetry properties are used to obtain the exact solutions or existence theorems in particular subspaces verifying such properties. Some basic concepts of differential geometry are involved, related with differentiable foliations of class \(C^2\). Suppose the solutions belongs to a space \(B\). The point is to split the solution into two components: one in an invariant subspace \(S\) and the second in its complement in \(B\). It is analyzed how the instability can be produced by the second component (related, for example, with the nonlinear part of the considered equations). The Poiseuille flow in an infinite pipe with circular cross-section is studied. The thermal flow between two horizontal coaxial cylinders kept at different temperatures is analyzed in the last part. The main mathematical tools are the Poincarè and Ladyzhenskaya inequalities and also some elements given in [\textit{G. P. Galdi} and \textit{B. Straughan}, Arch. Ration. Mech. Anal. 89, 211--228 (1985; Zbl 0622.76061)].