This paper gives another version of results due to Raugel and Sell, and similar results due to Moise, Temam and Ziane, that state the following: the solution of the Navier-Stokes equation on a thin three-dimensional domain with periodic boundary conditions has global regularity, as long as there is some control on the size of the initial data and the forcing term, where the control is larger than that obtainable via "small data" estimates. The approach taken is to consider the three-dimensional equation as a perturbation of the equation when the vector field does not depend upon the coordinate in the thin direction. 1. Introduction The celebrated Navier-Stokes equation is concerned with the velocity vector u on a domain# , describing the flow of an incompressible fluid. A famous unsolved problem is the following: if# is a nice enough 3-dimensional domain, and if the initial data is smooth, and the forcing term is uniformly smooth in time, then does it follow that the solution is smoo...

In view of the possibility that the 3D Navier-Stokes equations (NSE) might not always have regular solutions, we introduce an abstract framework for studying the asymptotic behavior of multi-valued dissipative evolutionary systems with respect to two topologies—weak and strong. Each such system possesses a global attractor in the weak topology, but not necessarily in the strong. In case the latter exists and is weakly closed, it coincides with the weak global attractor. We give a sufficient condition for the existence of the strong global attractor, which is verified for the 3D NSE when all solutions on the weak global attractor are strongly continuous. We also introduce and study a two-parameter family of models for the Navier-Stokes equations, with similar properties and open problems. These models always possess weak global attractors, but on some of them every solution blows up (in a norm stronger than the standard energy one) in finite time. Keywords: Navier-Stokes equations, global attractor, blow-up in finite time. 1

The motion of a fluid subject to Navier--Stokes equations with Coriolis force, to a Robin type traction condition at the surface and to a friction condition at the bottom is investigated. An asymptotic model is derived as the aspect ratio ffi = depth/width of the domain go to 0. When Reynolds number is not too large, this is mathematically justified and the 3D limit velocity is given in terms of wind, bathymetry, depth and of a 2D potential. Numerical simulation is carried out and the influence of traction condition reading is experienced. Keywords. Stationary Navier--Stokes equations, thin domain, Robin condition. AMS subjects classification. 35Q30, 35B40, 76D05. 1 Introduction The stationary motion of a fluid in a thin domain, satisfying Navier--Stokes equations with Coriolis force, a traction condition of Robin type at the surface and a friction condition at the bottom, is investigated. In view of the small aspect ratio ffi = depth=width of most lakes and seas, an asymptotic analy...

For the Euler equations in a thin domain Qε =Ω×(0,ε), Ω a rectangle in R 2, with initial data in (W 2,q (Qε)) 3, q>3, bounded uniformly in ε, the classical solution is shown to exist on a time interval (0,T(ε)), where T(ɛ) → + ∞ as ɛ → 0. We compare this solution with that of a system of limiting equations on Ω. 1

In this paper we study the behaviour of solutions of Navier--Stokes equations with adherence to the bottom and traction by wind at the surface when the aspect ratio ffi = depth=lenght of the domain tends to 0. Precisely we prove that when wind is moderate, a variational solution converges to the solution of a vertical diffusion model. This model is either quasistationary or nonstationary, according to the characteristic time. Keywords. Navier--Stokes equations, wind driven, Coriolis force, asymptotic model, shallow water. AMS subjects classification. 35Q30, 35B40, 76D05. 1 Introduction This paper is devoted to the solutions of Navier--Stokes equations with Coriolis force, traction by wind at the surface and adherence to the bottom in a thin domain. Their behaviour when the aspect ratio ffi = depth=lenght of the domain tends to 0 is investigated. At first we prove the existence of a variational solution for a fixed ffi, without regularity assumptions on the surface O. We only assume t...

. The motion of a fluid in a thin domain subject to wind traction at the surface and to slipping at the bottom is investigated. An asymptotic model is derived from Navier-- Stokes equations with Coriolis force as the aspect ratio ffi = depth/width of the domain go to 0, for not too large Reynolds number. Sur les 'ecoulements g'eophysiques engendr'es par le vent en l'absence de friction au fond R'esum'e. On s'int'eresse aux 'ecoulements en domaine mince soumis `a la traction du vent en surface et `a un glissement au fond. On d'eduit des 'equations de Navier--Stokes avec force de Coriolis un mod`ele asymptotique quand le rapport d'aspect ffi = hauteur/largeur du domaine tend vers 0, lorsque le nombre de Reynolds pas trop grand. Keywords. Stationary flow, thin domain, friction, Navier--Stokes. AMS subjects classification. 35Q30, 35B40, 76D05. On wind driven geophysical flows without bottom friction Didier BRESCH, Jacques SIMON Abstract. The motion of a fluid in a thin domain subjec...

Abstract. This study is motivated by problems arising in oceanic dynamics. Our focus is the Navier–Stokes equations in a three-dimensional domain Ωε, whose thickness is of order O(ε) as ε → 0, having non-trivial topography. The velocity field is subject to the Navier friction boundary conditions on the bottom and top boundaries of Ωε, and to the periodicity condition on its sides. Assume that the friction coefficients are of order O(ε 3/4) as ε → 0. It is shown that if the initial data, respectively, the body force, belongs to a large set of H 1 (Ωε), respectively, L 2 (Ωε), then the strong solution of the Navier–Stokes equations exists for all time. Our proofs rely on the study of the dependence of the Stokes operator on ε, and the non-linear estimate in which the contributions of the boundary integrals are non-trivial.

We consider the Navier–Stokes equations in a thin domain of which the top and bottom are not flat. The velocity fields are subject to the Navier conditions on those boundaries and the periodicity condition on the other sides of the domain. The model arises from studies of climate and oceanic flows. We show that the strong solutions exist for all time provided the initial data belong to a “large ” set in the Sobolev space H 1. Furthermore we show, for both the autonomous and the nonautonomous problems, the existence of a global attractor for the class of all globally-defined strong solutions. This attractor is proved to be also the global attractor for the Leray-Hopf weak solutions of the Navier–Stokes equations. One issue that arises here is a nontrivial contribution due to the boundary terms. We show how the boundary conditions imposed on the velocity fields affect the estimates of the Stokes operator and the (nonlinear) inertial term in the Navier–Stokes equations. This results in a new estimate of the trilinear

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ABSTRACT. We study the Navier-Stokes equations for incompressible fluids in a three-dimensional thin two-layer domain whose top, bottom and interface boundaries are not flat. In addition to the Navier friction boundary conditions on the top and bottom boundaries of the domain, and the periodicity condition on the sides, the fluid velocities are subject to an interface boundary condition which relates the normal stress of each fluid to the relative velocity between them on the common boundary. We prove that the strong solutions exist for all time if the initial data and body force, measured in relevant norms, are appropriately large as the domain becomes very thin. In our analysis, the interface boundary condition is interpreted as a variation of the Navier boundary conditions containing an interaction part. The effect of that interaction on the Stokes operator and the nonlinear term of the Navier-Stokes equations is expressed and carefully estimated in different ways in order to obtain suitable estimates. CONTENTS