Long-time asymptotics of the Navier-Stokes and vorticity equations on ℝ³

Abstract

We use the vorticity formulation to study the long-time behavior of solutions to the Navier-Stokes equation on R 3 . We assume that the initial vorticity is small and decays algebraically at infinity. After introducing self-similar variables, we compute the long-time asymptotics of the rescaled vorticity equation up to second order. Each term in the asymptotics is a self-similar divergence-free vector field with Gaussian decay at infinity, and the coefficients in the expansion can be determined by solving a finite system of ordinary differential equations. As a consequence of our results, we are able to characterize the set of solutions for which the velocity field satisfies u(·, t) L 2 = o(t −5/4 ) as t → +∞. In particular, we show that these solutions lie on a smooth invariant submanifold of codimension 11 in our function space.

Asymptotic behavior of the energy and pointwise estimates for solutions to the Navier-Stokes equations

“…This is obviously the case also for the condition ||u(t)|| 2 = o(t −(n+2)/4 ), but, on the other hand, (3.2) is not invariant. For a detailed discussion on this point, and the description of a method (very different from ours) to overcome this type of difficulties, we refer to [13], [14].…”

“…The bound on α is now known to be optimal: optimality was first discussed in [30] and, more recently, in [28], [13], [14] with different methods.…”

“…We refer to [13], [14] for an interesting geometrical insight of such rapidly decaying solutions. We also would like to observe that some of these results can be adapted in the case of flows in the half-space (see [11]).…”

Boundary characteristic point regularity for Navier–Stokes equations: Blow-up scaling and Petrovskii-type criterion (a formal approach)

“…Actually, the paper [24] deals with global asymptotics of NSEs solutions as t → +∞, where the adjoint operator B in (A.5) occurs. Since B is self-adjoint in L 2 ρ (R 3 ), many results from [25, Append.…”

“…9) div : Φ * 3 k → Φ * k−1 . We next define the corresponding solenoidal eigenspaces as follows: [24,25,26] and further references therein.…”