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

Abstract

We show that the solutions to the non-stationary Navier-Stokes equations in R d (d = 2, 3) which are left invariant under the action of discrete subgroups of the orthogonal group O(d) decay much faster as |x| → ∞ or t → ∞ than in generic case and we compute, for each subgroup, the precise decay rates in space-time of the velocity field.

On Two-Dimensional Navier-Stokes Flows with Rotational Symmetries

“…Contrary to the previous work [13], the basic existence result is proved with no restriction on the size of initial data. Our result extends that of [3] under di¤erent assumptions on the initial data. Unlike [3], the proofs are all carried out without using estimates for the associated vorticity transport equations.…”

“…Brandolese [3] proves (1.12) and (1.13) for r > 2 under di¤erent assumptions on the initial data. He first deduces some decay result for the associated vorticity and then applies the Biot-Savart law.…”

“…To this end, we follow [3] and employ the dihedral groups D m , m A N, m b 3 ( [7], [17]), of orthogonal transformations on R 2 . Consider the matrices A m ¼ cosð2p=mÞ Àsinð2p=mÞ sinð2p=mÞ cosð2p=mÞ…”

“…Proof. The idea below is due to [3]. Since dA n y ¼ jdet A n jdy ¼ dy, the D n -covariance of the vector field a gives ð að yÞdy ¼…”

D′alembert′s Paradox and the Integrability of Pressure for Non-Stationary Incompressible Euler Flows in a Two-Dimensional Exterior Domain

“…Here u = (u j ) 2 j =1 and p are, respectively, unknown velocity and pressure, a = (a j ) 2 j =1 is a given initial velocity and ν is the unit outward normal to ∂ . Let H m ( ), m ∈ N, be the usual Sobolev space with H 0 ( ) = L 2 ( ), and let L r σ ( ) = {u ∈ L r ( ) : ∇ · u = 0, ν · u| ∂ = 0}, 1 < r < ∞.…”