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16
Some results on the NavierStokes equations in thin 3D domains
 J Differential Equations
, 2001
"... Abstract. We consider the NavierStokes equations on thin 3D domains Qε = Ω×(0, ε), supplemented mainly with purely periodic boundary conditions or with periodic boundary conditions in the thin direction and homogeneous Dirichlet conditions on the lateral boundary. We prove global existence and uniq ..."
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Cited by 10 (1 self)
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Abstract. We consider the NavierStokes equations on thin 3D domains Qε = Ω×(0, ε), supplemented mainly with purely periodic boundary conditions or with periodic boundary conditions in the thin direction and homogeneous Dirichlet conditions on the lateral boundary. We prove global existence and uniqueness of solutions for initial data and forcing terms, which are larger and less regular than in previous works. An important tool in the proofs are some Sobolev embeddings into anisotropic Lptype spaces. As in [25], better results are proved in the purely periodic case, where the conservation of enstrophy property is used. For example, in the case of a vanishing forcing term, we prove global existence and uniqueness of solutions if ‖(I −M)u0 ‖ H 1/2 (Qε) exp(C −1 ε −1/s ‖Mu0 ‖ 2/s L 2 (Qε)) ≤ C for both boundary conditions or ‖Mu0 ‖ H 1 (Qε) ≤ Cε −β, ‖(Mu0)3 ‖ L 2 (Qε) ≤ Cε β, ‖(I − M)u0 ‖ H 1/2 (Qε) ≤ Cε 1/4−β/2 for purely periodic boundary conditions, where 1/2 ≤ s < 1 and 0 ≤ β ≤ 1/2 are arbitrary, C is a prescribed positive constant independent of ε and M denotes the average operator in the thin direction. We also give a new uniqueness criterium for weak Leray solutions.
Global regularity of the NavierStokes equation on thin three dimensional domains with periodic boundary conditions
 Electronic J. Diff. Eqns
, 1999
"... Abstract: 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 NavierStokes equation on a thin three dimensional domain with periodic boundary conditions has global regularity, as long as ..."
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Cited by 7 (0 self)
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Abstract: 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 NavierStokes 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. Keywords: NavierStokes equation, thin domain A.M.S. Classification (1991): Primary 35Q30, 76D05, Secondary 35B65. The celebrated NavierStokes equation is concerned with the velocity vector u on a domain Ω, describing the flow of an incompressible fluid. A famous unsolved problem is the following: if Ω 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 smooth for all time? What is known is that a weak solution exists, although it is not
On global attractors of the 3D NavierStokes equations
 J. Diff. eq
"... In view of the possibility that the 3D NavierStokes equations (NSE) might not always have regular solutions, we introduce an abstract framework for studying the asymptotic behavior of multivalued dissipative evolutionary systems with respect to two topologies—weak and strong. Each such system poss ..."
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Cited by 4 (4 self)
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In view of the possibility that the 3D NavierStokes equations (NSE) might not always have regular solutions, we introduce an abstract framework for studying the asymptotic behavior of multivalued 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 twoparameter family of models for the NavierStokes 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: NavierStokes equations, global attractor, blowup in finite time. 1
On the random kickforced 3D NavierStokes equations in a thin domain, Archive for Rational Mechanics and Analysis, to appear; see also mp arc
"... We consider the NavierStokes equations in the thin 3D domain T 2 × (0, ε), where T 2 is a twodimensional torus. The equation is perturbed by a nondegenerate random kickforce. We establish that, firstly, when ε ≪ 1 the equation has a unique stationary measure and, secondly, after averaging in the ..."
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Cited by 3 (2 self)
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We consider the NavierStokes equations in the thin 3D domain T 2 × (0, ε), where T 2 is a twodimensional torus. The equation is perturbed by a nondegenerate random kickforce. We establish that, firstly, when ε ≪ 1 the equation has a unique stationary measure and, secondly, after averaging in the thin direction this measure converges (as ε → 0) to a unique stationary measure for the NavierStokes equation on T 2. Thus, the latter describes asymptotic in time and limiting in ε statistical properties of the NavierStokes flow in the narrow 3D domain. MSC: primary 35Q30; secondary 35R60, 76D06, 76F55. Keywords: NavierStokes equations; thin domains; random kicks, convergence in law, stationary measure. 1
Navier–Stokes equations with Navier Boundary Conditions in Nearly Flat Domains
, 2008
"... 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. ..."
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Cited by 3 (2 self)
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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 H¹. Furthermore we show, for both the autonomous and the nonautonomous problems, the existence of a global attractor for the class of all globallydefined strong solutions. This attractor is proved to be also the global attractor for the LerayHopf 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
Stochastic 3D NavierStokes equations in a thin domain and its αapproximation
, 2007
"... In the thin domain Oε = T 2 × (0, ε), where T 2 is a twodimensional torus, we consider the 3D NavierStokes equations, perturbed by a white in time random force, and the Leray αapproximation for this system. We study ergodic properties of these models and their connection with the corresponding 2D ..."
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Cited by 2 (1 self)
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In the thin domain Oε = T 2 × (0, ε), where T 2 is a twodimensional torus, we consider the 3D NavierStokes equations, perturbed by a white in time random force, and the Leray αapproximation for this system. We study ergodic properties of these models and their connection with the corresponding 2D models in the limit ε → 0. In particular, under natural conditions concerning the noise we show that in some rigorous sense the 2D stationary measure µ comprises asymptotical in time statistical properties of solutions for the 3D NavierStokes equations in Oε, when ε ≪ 1.
On the Effect of Friction on Wind Driven Shallow Lakes
, 1999
"... The motion of a fluid subject to NavierStokes 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 ..."
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Cited by 1 (1 self)
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The motion of a fluid subject to NavierStokes 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 NavierStokes equations, thin domain, Robin condition. AMS subjects classification. 35Q30, 35B40, 76D05. 1 Introduction The stationary motion of a fluid in a thin domain, satisfying NavierStokes 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...
The Euler Equations on Thin Domains
"... 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 e ..."
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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
Nonstationary Models for Shallow Lakes
, 1999
"... In this paper we study the behaviour of solutions of NavierStokes 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 sol ..."
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In this paper we study the behaviour of solutions of NavierStokes 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. NavierStokes 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 NavierStokes 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...
Blaise Pascal
, 1999
"... . 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 nu ..."
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. 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 NavierStokes 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, NavierStokes. 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...