Computer-assisted proofs of the existence of a symmetry-breaking bifurcation point for the Kolmogorov problem

Abstract

We propose a computer-assisted method to prove the existence of a symmetry-breaking bifurcation point for the Kolmogorov problem. First, we numerically show that a symmetry-breaking bifurcation point exists. Then, according to the symmetric property, we define a symmetric operator. Using this operator, we divide the space into a symmetric space and an antisymmetric space. Then, considering the Reynolds number as a variable, we construct an extended system. We confirm the existence of the symmetry-breaking bifurcation point by computer-assisted proofs of the extended system that satisfies both conditions of a bifurcation theorem. The first condition is that the system has an isolated solution and the second is that a linearized operator is bijective. We numerically construct a set containing solutions that satisfy the hypothesis of Banach’s fixed-point theorem in a certain Sobolev space and thus the first condition is satisfied. The second condition is equivalent to an equation having the unique trivial solution zero. We prove that this condition is equivalent to an inequality.