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Considering the Kolmogorov flow, an approximate nontrivial solution is easily obtained using a numerical method. A numerical verification method then allows us to enclose a nontrivial solution near th...e approximate one. Our aim is to prove the stability of the verified solution. This stability can be transformed to the sign of the real parts of the eigenvalues for a linearized operator at the solution; that is, if the real part is negative, then the solution is stable. Numerically, we discover that the eigenvalues of the operator linearized at the approximate solution satisfy the principle of the exchange of stabilities (PES), which implies that the imaginary part of the first unstable eigenvalue of the linearized system is equal to zero. Therefore, we assume that our linearized system at the verified solution satisfies PES. Then, we implement an eigenvalue excluding method for the real axis. For the operator linearized at the verified solution, we prove an eigenvalue excluding theorem. By the theorem, we obtain under which conditions and in which disks there will be no eigenvalues. It is difficult to find those disks directly. However, for the operator linearized at the approximate solution, it is possible to construct various computable criteria that allow us to use a computer program to exclude the eigenvalues. Then combining the verified results of the solution, we can obtain the area where the eigenvalues for the operator linearized at the verified solution are not present. Finally, we prove the stability of two nontrivial solutions by our method.続きを見る
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