Abstract. We consider the incompressible Euler and Navier-Stokes equations in a three-dimensional moving thin domain. Under the assumption that the moving thin domain degenerates into a two-dimensional moving closed surface as the width of the thin domain goes to zero, we give a heuristic derivation of singular limit equations on the degenerate moving surface of the Euler and Navier-Stokes equations in the moving thin domain and investigate relations between their energy structures. We also compare the limit equations with the Euler and Navier-Stokes equations on a stationary manifold, which are described in terms of the Levi-Civita connection. IntroductionFluid flows in a thin domain appear in many problems of natural sciences, e.g. ocean dynamics, geophysical fluid dynamics, and fluid flows in cell membranes. In the study of the incompressible Navier-Stokes equations in a three-dimensional thin domain mathematical researchers are mainly interested in global existence of a strong solution for large data since a three-dimensional thin domain with sufficiently small width can be considered "almost two-dimensional." It is also important to investigate the behavior of a solution as the width of a thin domain goes to zero. We may naturally ask whether we can derive limit equations as a thin domain degenerates into a two-dimensional set and compare properties of solutions to the original three-dimensional equations and the corresponding two-dimensional limit equations. There are several works studying such problems with a three-dimensional flat thin domain [15,16,29,33] of the formfor small ε > 0, where ω is a two-dimensional domain and g 0 and g 1 are functions on ω, and a three-dimensional thin spherical domain [34] which is a region between two concentric spheres of near radii. (We also refer to [28] for the strategy of analysis of the Euler equations in a flat and spherical thin domain and its limit equations.) However, mathematical studies of an incompressible fluid in a thin domain have not been done in the case where a thin domain and its degenerate set have more complicated geometric structures. (See [27] for the mathematical analysis of a reaction-diffusion equation in a thin domain degenerating into a lower dimensional manifold.) In this paper we are concerned with the incompressible Euler and Navier-Stokes equations in a three-dimensional thin domain that moves in time. The purpose of this paper is to give a heuristic derivation of singular limits of these equations as a moving thin domain degenerates into a two-dimensional moving closed surface. We also investigate relations between the energy structures of the incompressible fluid 2010 Mathematics Subject Classification. Primary: 35Q35, 35R01, 76M45; Secondary: 76A20.

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We consider the Navier-Stokes equations on the two-dimensional unit sphere and study the linear stability of the two-jet Kolmogorov type flow which is a stationary solution given by the zonal spherical harmonic function of degree two. We prove the linear stability of the two-jet Kolmogorov type flow for an arbitrary viscosity coefficient by showing the exponential decay of a solution to the linearized equation towards an equilibrium which grows as the viscosity coefficient tends to zero. Moreover, we show that the perturbation operator appearing in the linearized equation does not have nonzero eigenvalues by using the mixing property of the perturbation operator expressed by a recurrence relation for the spherical harmonics. As an application of this result, we get the enhanced dissipation for a solution to the linearized equation rescaled in time in the sense that the solution decays arbitrarily fast as the viscosity coefficient tends to zero.

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