Lower bounds on blow up solutions of the three-dimensional Navier–Stokes equations in homogeneous Sobolev spaces

Abstract

Suppose that u(t) is a solution of the three-dimensional Navier-Stokes equations, either on the whole space or with periodic boundary conditions, that has a singularity at time T. In this paper we show that the norm of u(T − t) in the homogeneous Sobolev spaceḢ s must be bounded below by c s t

Properties at potential blow-up times for Navier-Stokes

“…This issue is well studied in the literature, see e.g. [3,11,12,13,14,15,16,17,18,19], but the results are somewhat scattered. We will review some results that we consider to be very important and will also derive lower bounds for some blow-up rates.…”

“…The estimate (4.20) has been recently shown in [18] to hold for q = 2 as well, but its validity for arbitrary q ≥ 3/2 seems to be still open. The general fact that the norms…”

“…For additional lower bound estimates and results concerning other blow-up quantities, the reader is referred to [11,18,38,47,48].…”

Characterization of blowup for the Navier-Stokes equations using vector potentials

“…18. Lots of progress has been made since then, including 2,3,[5][6][7][8][9][10][11][12]15,19,21,[23][24][25][26]28 and various kinds of blowup or regularity criteria have been developed. We define the vector potential A by u = ∇ × A in three dimensions, where ∇ · A = 0 and the stream function in two dimensions by u = (∂ 2 ψ, −∂ 1 ψ).…”

Study of the Navier–Stokes regularity problem with critical norms

“…We study the basic problems of regularity of the Navier-Stokes equations. The blowup criteria on the basis of the critical H 1/2 -norm, is bounded from above by a logarithmic function, Robinson, Sadowski and Silva (2012). Assuming that the Cauchy-Schwarz inequality for the H 1/2 -norm is not an overestimate, we replace it by a square-root of a product of the energy and the enstrophy.…”