AUTHORS: Atefeh Hasan-Zadeh, Mohammad Mohammadi Khanaposhtani
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ABSTRACT: Inertial manifolds of Navier-Stokes equations have been calculated approximately up to now. In this paper, drawing upon advanced ingredients of differential geometry and Lie groups a novel methodology is presented for finding the inertial manifolds of ( ) +12 -dimensional Navier-Stokes equation. It has been shown that the geometric notions about Lie groups and Lie algebras such as transformation groups, one-parameter groups, integral submanifolds, adjoint representations, group-invariant solutions and optimal systems not only cover all of the properties of inertial manifolds, but also result to the exact decomposition of the velocity field of the flow of Navier-Stokes equation by proposing the coordinate chart for it. In this way, the new procedure outperforms the numerical estimation methods by providing the analytic solution of the inertial manifolds. Also, the proposed methodology can be applied to the general problems by searching the optimal systems of them. Furthermore, this geometric approach results to the reduction theory which transforms these partial differential equations into a system of differential equations with fewer variables.
KEYWORDS: Navier-Stokes equation, inertial manifold, Lie algebra, optimal system, invariant solution, Frobenius' theorem
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