Asymptotic Analysis - Volume 15, issue 2

Authors: Babin, A. | Mahalov, A. | Nicolaenko, B.

Article Type: Research Article

Abstract: We consider 3D Euler and Navier–Stokes equations describing dynamics of uniformly rotating fluids. Periodic (as well as zero vertical flux) boundary conditions are imposed, the ratios of domain periods are assumed to be generic (non-resonant). We show that solutions of 3D Euler/Navier–Stokes equations can be decomposed as U(t,x1 ,x2 ,x3 )=Ũ(t,x1 ,x2 ) +V(t,x1 ,x2 ,x3 )+r, where Ũ is a solution of the 2D Euler/Navier–Stokes system with vertically averaged initial data (axis of rotation is taken along the vertical e3 ). The vector field V(t,x1 ,x2 ,x3 ) is exactly solved in terms of the phases Ωt, τ1 (t) …and τ2 (t). The phases τ1 (t) and τ2 (t) explicitly expressed in terms of vertically averaged vertical vorticity curl Ū(t)·e3 and velocity Ū3 (t). The remainder r is uniformly estimated from above by a majorant of order a3 /Ω, a3 is the vertical aspect ratio (shallowness) and Ω is non-dimensional rotation parameter based on horizontal scales. The resolution of resonances and a non-standard small divisor problem for 3D rotating Euler are the basis for error estimates. Contribution of 3-wave resonances is estimated in terms of the measure of almost resonant aspect ratios. Global solvability of the limit equations and estimates of the error r are used to prove existence on a long time interval T* of regular solutions to 3D Euler equations (T* →+∞, as 1/Ω→0); and existence on infinite time interval of regular solutions to 3D Navier–Stokes equations with smooth arbitrary initial data in the case of small 1/Ω. Show more

DOI: 10.3233/ASY-1997-15201

Citation: Asymptotic Analysis, vol. 15, no. 2, pp. 103-150, 1997

Authors: Didelot, S. | Frank, L.S. | Maigrot, L.

Article Type: Research Article

Abstract: The method of constructive reduction of elliptic and coercive singular perturbations to regular ones, developed in [2–7] is applied to so-called bisingular singularly perturbed boundary value problems (see, for instance, [8]). The reduction procedure allows to derive asymptotic formulae for the solutions of these problems, without going through the matching of different locally formally derived asymptotic expansions (see [8,9,11]). The advantage of using the reduction procedure consists not only in the globality of asymptotic formulae thus obtained, but also in a simple proof of their asymptotic convergence as a direct consequence of the central reduction to regular perturbations result.

DOI: 10.3233/ASY-1997-15202

Citation: Asymptotic Analysis, vol. 15, no. 2, pp. 151-181, 1997

Authors: Abddaimi, Y. | Michaille, G. | Licht, C.

Article Type: Research Article

Abstract: We establish the epiconvergence (or Γ-convergence) for the strong topology of L1 (O,RN ) of a sequence of random integral functionals ∫O f(ω)(x/ε,∇u(x)) dx defined in the space W1,1 (O,RN ). Our method uses an improvement of a convergence result concerning subadditive processes and a recent technique of blow-up combined with the rank one property of the singular measure Ds u.

Keywords: Homogenization, epiconvergence, stochastic processes, ergodic theory

DOI: 10.3233/ASY-1997-15203

Citation: Asymptotic Analysis, vol. 15, no. 2, pp. 183-202, 1997