TY - JOUR

T1 - Computer-assisted proofs of the existence of a symmetry-breaking bifurcation point for the Kolmogorov problem

AU - Cai, Shuting

AU - Watanabe, Yoshitaka

N1 - Funding Information:
This work was supported by Grants-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology of Japan (No. 15H03637 ) and the Japan Science and Technology Agency, CREST (No. JPMJCR14D4 ). The computations were mainly carried out using the computer facilities at the Research Institute for Information Technology, Kyushu University, Japan.
Funding Information:
This work was supported by Grants-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology of Japan (No. 15H03637) and the Japan Science and Technology Agency, CREST (No. JPMJCR14D4). The computations were mainly carried out using the computer facilities at the Research Institute for Information Technology, Kyushu University, Japan. The authors heartily thank the two anonymous referees for their thorough reading and valuable comments.
Publisher Copyright:
© 2021 Elsevier B.V.

PY - 2021/10/15

Y1 - 2021/10/15

N2 - 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.

AB - 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.

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U2 - 10.1016/j.cam.2021.113603

DO - 10.1016/j.cam.2021.113603

M3 - Article

AN - SCOPUS:85104656765

SN - 0377-0427

VL - 395

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

M1 - 113603

ER -