Decomposition of solenoidal fields into poloidal fields, toroidal fields and the mean flow. Applications to the boussinesq-equations

About the Approximation of Unsteady Flows by the Eigenfunctions of a Certain Differential Operator

“…Now the typical linear Galerkin method using divergence -free basis functions proceeds by identifying a sequence of linearly independent real functions {u k } ∞ k=1 with u k ∈ H ∩ H 1 0 , such that the linear combinations of these functions are dense in H. These functions might be eigenfunctions of S or L, or defined as in [3,17,18,19,20,28], or they might be finite elements. One multiplies Equation (1.1) by u l scalarly and solves the system of ordinary differential equations ∩ H, which we will assume from now on, we can rephrase this equation as follows.…”

Asymptotic stability of higher order norms in terms of asymptotic energy stability for viscous incompressible fluid flows heated from below

“…The solenoidal vector fields u being periodic in (x, y) e R 2, say with respeet to a rectangle Ÿ = (_~, ~) • (_~, ~) are decomposed into (k = (0, 0, i)T) u = curl curl ~_k + curl r + f with functions ~, r having the same periodicity as u and vanishing mean value over Ÿ The mean flow f depends on z only with first and second components fl, f2 anda constant third component (cf. [4]). Let ~58 be any sufficiently smooth steady solution of the inhomogeneous Boussinesq-equations.…”

“…Otherwise the boundary values of ~~ are not specified. The problem (1.11) can be dealt with in the same way as we did for (1.7) with ~8 -0 in [8], [4], and with the same difficulties. In particular the occurrence of highest order derivatives in the 91 in the first row enforces a complicated iteration procedure ; "highest order" refers to _f and the z-derivatives of r As for (1.11) we can therefore stick to the saune regulaxity and existence classes of solutions.…”

“…The 2 2highest order derivatives of Uz = -(O z + Oy)~ ate isolated in a single equation, the nonlinearity AA(qs, qs), and Ad(~, ~5s) and Ad(q5, qs) are almost local. This material was discussed in detail in[4].A vector field _u or f is usually written as a column, i.e. u = (Ul,U2,' 91 T :(Ux, Uy, Uz) T, s : (fl, f2, f3) T : (fx, fy, fz) T with the symbol 9 T for transposition.Correspondingly we sometimes write (x~, x2, x3) for (x, y, z).…”

About the Approximation of Steady Viscous Flows Using the Eigenfunctions of a Certain Differential Operator

“…In addition, the method based on the eigenfunctions of the Stokes operator seems to encounter other difficulties (see [15]). The method based on a representation given in [5] [6] [7] [23] [28], which was proved to hold for a spherical shell in [5] and for a periodic layer between two parallel planes in [28], and used numerically in [6] [9] [26] [27] seems to lead to convergence in H 3 . 5 -e .…”