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In this remark, we consider regularity criterion for weak solutions to the 3d incompressible Navier–Stokes equations via pressure. It is proved that if the corressponding pressure P satisfies ∇ h ˜ P ∈ L β (0 , T ; L p (R x 1 ; L q (R x 2 ; L r (R x 3)))) with 2 β + 1 p + 1 q + 1 r = 3 , 6 5 < p , q , r ≤ 3 and 2 − (1 p + 1 q + 1 r) ≥ 0 , then the weak solution remains smooth on (0 , T ] . Here, ∇ h ˜ = (0 , ∂ 2 , ∂ 3) . [ABSTRACT FROM AUTHOR]

This work establishes a new regularity criterion for the 3D incompressible MHD equations in term of one directional derivative of the pressure (i.e., ∂ 3 P ) on framework of the anisotropic Lebesgue spaces. More precisely, it is proved that for T > 0 , if ∂ 3 P ∈ L β (0 , T ; L α (R x 1 x 2 2 ; L γ (R x 3))) with 2 β + 1 γ + 2 α = k ∈ [ 2 , 3) and 3 k ≤ γ ≤ α ≤ 1 k - 2 , then the corresponding solution (u, b) to the 3D MHD equations is regular on [0, T]. [ABSTRACT FROM AUTHOR]

This work establishes a new logarithmical improved regularity criterion for the 3D Boussinesq equations in terms of one directional derivative of the pressure (i.e., ∂ 3 P ) on framework of the anisotropic Lebesgue spaces. More precisely, it is proved that if ∫ 0 T ‖ ∂ 3 P (· , t) ‖ L x 3 γ L x 1 x 2 α β 1 + ln (1 + ‖ θ (· , t) ‖ L 4) d t < ∞ , where 2 β + 1 γ + 2 α = k ∈ [ 2 , 3) and 3 k ≤ γ ≤ α < 1 k - 2 , for some T > 0 , then the corresponding solution (u , θ) to the 3D Boussinesq equations is regular on [0, T]. [ABSTRACT FROM AUTHOR]

In this paper, we consider some sufficient conditions for the breakdown of local smooth solutions to the Cauchy problem of the 3D Navier-Stokes/Poisson-Nernst-Planck system modeling electro-diffusion in terms of pressure (or gradient of pressure or one directional derivative of pressure) in the framework of the anisotropic Lebesgue spaces. Precisely, let T be the maximum existence time of local smooth solution. Then if T<+∞, we have ∫0TPLx1pLx2qLx3rβdt=+∞,where 2β+1p+1q+1r=2, 2≤p,q,r≤∞ and 1-(1p+1q+1r)≥0, and ∫0T∇PLx1pLx2qLx3rβdt=+∞,where 2β+1p+1q+1r=3, 1≤p,q,r≤∞ and 2-(1p+1q+1r)≥0, and ∫0T‖∂3P‖Lx3γLx1x2αβdt=+∞,where 2β+1γ+2α=k∈[2,3) and 3k≤γ≤α<1k-2. These results are even new for the 3D incompressible Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]