Atomic decomposition for the vorticity of a viscous flow in the whole space

Abstract

We show that the vorticity of a viscous flow in R 3 admits an atomic decomposition of the form ω(, with localized and oscillating building blocks ω k , if such a property is satisfied at the beginning of the evolution. We also study the long time behavior of an isolated coherent structure and the special behavior of flows with highly oscillating vorticities.

Space-time decay of Navier–Stokes flows invariant under rotations

“…Note that the decay profiles of ω and Ω are the same which can be obtained for the solutions of the homogeneous heat equations e t∆ ω 0 (x) and e t∆ Ω 0 (x), respectively. We refer to [6] for a proof of (13) (the proof of (11) is identical). The decay of the vorticity in the L p -norm are formally a consequence of (11) and (13), respectively if d = 2 or 3.…”

“…The decay of the vorticity in the L p -norm are formally a consequence of (11) and (13), respectively if d = 2 or 3. Estimates (12) and (14), however, can be proved with straightforward adaptations of the arguments of [11]- [6], or [16]- [17] (see also [8]).…”

“…The condition on the moments of u is easily seen via the Fourier transform and the Taylor formula (see e.g. [6] and [17] for this type of calculation). •…”

Poisson kernels and sparse wavelet expansions

“…where (x m ) is a sequence of points in R n and u m (·, t) are L ∞ functions decaying at infinity as |x| −(n+1) , holds true (see [1]). Expanding u(·, t) ∈ p>0Ḃ n/p,p p (R n ) by means of dilated-translated Poisson kernels (which have the "right" decay rate) is therefore more meaningful, from the physical point of view, than expressing the velocity field as a series of too localized functions, such as gaussians or wavelets.…”

“…the motion of a viscous incompressible fluid filling the whole space R n and not submitted to the action of external forces. If u(x, t) denotes the velocity of a fluid particle at time t, then under suitable assumptions on the vorticity of the flow, it can be shown that u(·, t) ∈ p>0Ḃ n/p,p p (R n ), uniformly in time (see [1]). For an application of sparse wavelet expansions to numerical simulations of flows we refer to [15].…”