ε-Regularity criteria in anisotropic Lebesgue spaces and Leray’s self-similar solutions to the 3D Navier–Stokes equations

In this paper, we establish some ε -regularity criteria in anisotropic Lebesgue spaces for suitable weak solutions to the 3D Navier–Stokes equations as follows: 0.1 lim sup ϱ → 0 ϱ 1 - 2 p - ∑ j = 1 3 1 q j ‖ u ‖ L t p L x q → ( Q ( ϱ ) ) ≤ ε , 2 p + ∑ j = 1 3 1 q j ≤ 2 with q j > 1 ; sup - 1 ≤ t ≤ 0 ‖ u ‖ L q → ( B ( 1 ) ) ≤ ε , 1 q 1 + 1 q 2 + 1 q 3 < 2 with 1 < q j < ∞ ; ‖ u ‖ L t p L x q → ( Q ( 1 ) ) + ‖ Π ‖ L 1 ( Q ( 1 ) ) ≤ ε , 2 p + ∑ j = 1 3 1 q j < 2 with 1 < q j < ∞ , which extends the previous results in Caffarelli et al. (Commun Pure Appl Math 35:771–831, 1982), Choi and Vasseur (Ann Inst H Poincaré Anal Non Linéaire 31:899–945, 2014), Gustafson et al. (Commun Math Phys 273:161–176, 2007), Guevara and Phuc Calc Var 56:68, 2017), He et al. (J Nonlinear Sci 29:2681–2698, 2019), Tian and Xin (Commun Anal Geom 7:221–257, 1999) and Wolf (Ann Univ Ferrara 61:149–171, 2015). As an application, in the spirit of Chae and Wolf (Arch Ration Mech Anal 225:549–572, 2017), we prove that there does not exist a nontrivial Leray’s backward self-similar solution with profiles in L p → ( R 3 ) with 1 p 1 + 1 p 2 + 1 p 3 < 2 . This generalizes the corresponding results of Chae and Wolf (Arch Ration Mech Anal 225:549–572, 2017), Guevara and Phuc (SIAM J Math Anal 50:541–556, 2017), Nečas et al. (Acta Math 176, 283–294, 1996) and Tsai (Arch Ration Mech Anal 143(1):29–51, 1998).