Computer assisted proofs of bifurcating solutions for nonlinear heat convection problems

In previous works (Nakao et al., Reliab. Comput., 9(5):359-372, 2003; Watanabe et al., J. Math. Fluid Mech., 6(1):1-20, 2004), the authors considered the numerical verification method of solutions for two-dimensional heat convection problems known as Rayleigh-Bénard problem. In the present paper, to make the arguments self-contained, we first summarize these results including the basic formulation of the problem with numerical examples. Next, we will give a method to verify the bifurcation point itself, which should be an important information to clarify the global bifurcation structure, and show a numerical example. Finally, an extension to the three dimensional case will be described.