Addendum to the Paper ”two-dimensional Infinite Prandtl Number Convection: Structure of Bifurcated Solutions, J.

The main objective of this addendum to the mentioned article [49] by Park is to provide some remarks on bifurcation theories for nonlinear partial differential equations (PDE) and their applications to fluid dynamics problems. We only wish to comment and list some related literatures, without any intention to provide a complete survey. For steady state PDE bifurcation problems, the often used classical bifurcation methods include 1) the Lyapunov-Schmidt procedure, which reduces the PDE problem to a finite dimensional algebraic system, 2) the Krasnoselskii theorem for bifurcations crossing an eigenvalue of odd algebraic multiciplicity [28] , 3) the Krasnoselskii theorem for potential operators, 4) the Rabinowitz global bifurcation theorem [51], 5) Crandall and Rabinowitz theorem for bifurcations crossing a simple eigenvalue [11], and 6) bifurcation from higher-order terms, regardless of the multiplicity of the eigenvalues [31, 32]. We also refer the interested readers to, among many others, [47, 8, 20, 21, 35, 42] for more comprehensive discussions. Nirenberg have a beautiful survey paper [46] on topological and variational methods for nonlinear problems, which has influenced a whole generation of nonlinear analysts. The Hopf bifurcation, also called Poincaré-Andronov-Hopf bifurcation, was independently studied and discovered by Andronov in 1929 and Hopf in 1942 and Poincaré in 1892 for ordinary differential equations. In particular, in his paper [24], Hopf also indicated the possible application of the Hopf bifurcation theorem to bifurcation of time periodic solutions for the Navier-Stokes equations. The Hopf bifurcation was generalized to infinite dimensional setting for PDEs by Crandall and Rabinowitz [12], Marsden and McCracken [44], and Henry [23]. We mention in particular the last two references using the center manifold reduction procedure to reduce the problem to a finite dimensional problem.