Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations

Abstract

Abstract. We study locally self-similar solutions of the three dimensional incompressible Navier-Stokes equations. The locally self-similar solutions we consider here are different from the global self-similar solutions. The self-similar scaling is only valid in an inner core region that shrinks to a point dynamically as the time, t, approaches a possible singularity time, T . The solution outside the inner core region is assumed to be regular, but it does not satisfy selfsimilar scaling. Under the assumption that the dynamically rescaled velocity profile converges to a limiting profile as t → T in L p for some p ∈ (3, ∞), we prove that such a locally self-similar blow-up is not possible. We also obtain a simple but useful non-blowup criterion for the 3D Euler equations.

Late formation of singularities in solutions to the Navier–Stokes equations

“…then U is a steady solution to the Leray equations and hence U 0 º by [26]. This is the argument developed in [2,16]. Note that weak solutions to the steady Leray equations are known to be actually smooth [26].…”

“…As we mentioned above, self-similar blowup [26,39] and asymptotically self-similar blowup [2,16] have been excluded for the first singularity. See also [3,4] for related works.…”

“…Substantial progress has been made recently in the understanding of the regularity properties of the incompressible Navier-Stokes equations. Ruling out self-similar blowup and improving blowup criteria with critical norms are just a few examples of recent results [2,11,16,26,34,39]. In this paper we study dynamically scaled version of the Navier-Stokes equations, called non-steady Leray equations in detail.…”

Cole–Hopf-Feynman–Kac formula and quasi-invariance for Navier–Stokes equations

“…Assuming that a solution to the Navier-Stokes equations blows up at t = t * , we apply the dynamic scaling transformations e.g. [4,14,20,22]…”

“…The above comparison is based on the presence or absence of one multiplicative factor G a . It is a 'good' factor in that it restores smoothness which is otherwise broken at t = 1/2a; if G a is neglected in (14), the solution becomes short-lived. On the other hand, if we insert G a , which is valid as a martingale in t < √ 2/a, into (13), we obtain (22) below.…”