Regularity criterion via the pressure on weak solutions to the 3D Navier-Stokes equations

Abstract

Abstract. We consider the regularity of weak solutions to the Navier-Stokes equations in R 3 . Let u be a Leray-Hopf weak solution. It is proved that u becomes a regular solution if the pressure p ∈ L 1 (0, T ;Ḃ 0 ∞,∞ ).

On blow-up "twistors" for the Navier--Stokes equations in $R^3$: a view from reaction-diffusion theory

“…The condition (2.17) is consistent with Leray-Prodi-Serrin-Ladyzhenskaya regularity L p,q criteria and other more recent researches; see key references, history, details, and results concerning this huge existence-regularity-blow-up business around the NSEs in [53,63,134,148,182,192,202,219]; [57] represents a modern panorama of such studies, which also commented that Ohyama's result in 1960 [166] was obtained before Serrin's one in 1962 [194]. (2.17) is also associated with Kato's class of unique mild solutions (in R N ), [115]; see details and key references in [29,63,216].…”

“…1, § 6] for a discussion); q.v. e.g., [182] and references therein. (2.2) is also a crucial identity for general turbulence theory.…”

On global solutions and blow-up for Kuramoto–Sivashinsky-type models, and well-posed Burnett equations

“…The nonblow-up also is proved in the well-known critical case p = N = 2. The condition (1.17) is consistent with Leray-Prodi-Serrin-Ladyzhenskaya regularity L p,q criteria and other more recent results; see key references, history, details, and results in recent papers [65,20,69,25]. In addition, (1.17) re-derives Kato's class of unique mild solutions in R N , [42]; see details and key references in [25,77].…”

“…Actually, the estimate in (6.8) is the energy inequality for Leray-Hopf weak solutions of (1.6); see e.g., [65] and references therein. Nevertheless, an L 2 -bound is not sufficient to control non-blowing up property of solutions, and this is the origin of extensive mathematical research in the last fifty years.…”

New Regularity Criteria for the Navier-Stokes Equations in Terms of Pressure

“…Some of these conditions have also been further generalized using weaker norms. See for example, [8] , [20], [15], [29], [1], [6], [26], [4], [11], [2], [21], [36], [3], [5], [35], [34], [13], [23], [7], [19] and [24] for several results of this form.…”