International Symposium on Nonlinear Theory and its Applications
Some computer assisted proofs on the bifurcation structure of solutions for heat convection problems
M. T. Nakao, Y. Watanabe, N. Yamamoto, T. Nishida, M-N. Kim,
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In this paper, we present several results on cmputer assisted approaches for solutions of the two-dimensional Rayleigh-Benard convection problems. First, we will describe on a basic concept of our numerical verification method to prove the exsistence of the steady-state solutions based on the infinite dimensional fixed-point theorem using Newtonlike operator with the spectral approximation and the constructive error estimates. Next, we show some verification examples of several exact non-trivial solutions for the given Prandtl and Rayleigh numbers. Furthermore, a computer assisted proof of the existence for a symmetry breaking bifurcation point will be presented, which should be an important information to clarify the global bifurcation structure. We will also consider the extension of these results to the three dimensional problems.