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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]

The aim of this paper is to establish the regularity criterion of weak solutions to the 3D micropolar fluid equations by one directional derivative of the pressure in anisotropic Lebesgue spaces. We improve the regularity criterion for weak solutions previously given by Jia, Zhang and Dong in [21].

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]

Abstract This work establishes a new regularity criterion for the 3D incompressible magneto-hydrodynamical (MHD) equations in terms of one directional derivative of the velocity (i.e., ∂ 3 u) on framework of the anisotropic Lebesgue spaces. More precisely, it is proved that if ∂ 3 u satisfies ∫ 0 T ‖ ‖ ∂ 3 u (τ) ‖ L x 3 α ‖ L x 1 x 2 β q d τ < + ∞ , where 2 q + 1 α + 2 β = k ∈ [ 1 , 3 2 ] and 3 k < α ≤ β ≤ 1 k − 1 , for some T > 0 , then the corresponding solution (u , b) to the 3D MHD equations is regular on (0 , T) × R 3. [ABSTRACT FROM AUTHOR]

Abstract This paper establishes two logarithmically improved regularity criteria for the Cauchy problem of the 3D Boussinesq equations in termsof the pressure on framework of the anisotropic Lebesgue spaces. More precisely, it is proved that if ∫ 0 T ‖ ‖ ‖ P ‖ L x 1 p ‖ L x 2 q ‖ L x 3 r β 1 + ln (1 + ‖ θ (⋅ , t) ‖ L 4 ) d t < ∞ , where 2 β + 1 p + 1 q + 1 r = 2 and 2 ≤ p , q , r ≤ ∞ , 1 − (1 p + 1 q + 1 r) ≥ 0 , or ∫ 0 T ‖ ‖ ‖ ∇ P ‖ L x 1 p ‖ L x 2 q ‖ L x 3 r β 1 + ln (1 + ‖ θ (⋅ , t) ‖ L 4 ) d t < ∞ , where 2 β + 1 p + 1 q + 1 r = 3 and 1 ≤ p , q , r ≤ ∞ , 1 − (1 2 p + 1 2 q + 1 2 r) ≥ 0 , for some T > 0 , then the corresponding weak solution (u , θ) to the 3D Boussinesq equations is regular on [ 0 , T ]. [ABSTRACT FROM AUTHOR]

In this paper, we concern the 3D nematic liquid crystal equations and prove three almost Serrin-type blow-up criteria for the breakdown of local in time smooth solutions in terms of pressure and gradient of the orientation field. More precisely, let T ∗ be the maximal time of the local smooth solution, then T ∗ < + ∞ if and only if ∫ 0 T ∗ ‖ ‖ ‖ P ( ⋅ , t ) ‖ L x 1 p ‖ L x 2 q ‖ L x 3 r β + ‖∇d(⋅,t)‖ L 4 8 d t = ∞ , with 2 β + 1 p + 1 q + 1 r = 2 and 2 ≤ p , q , r ≤ ∞ , 1 − ( 1 p + 1 q + 1 r ) ≥ 0 , and ∫ 0 T ∗ ‖ ‖ ‖ ∇ P ( ⋅ , t ) ‖ L x 1 p ‖ L x 2 q ‖ L x 3 r β + ‖ ∇ d ( ⋅ , t ) ‖ L 4 8 d t = ∞ , with 2 β + 1 p + 1 q + 1 r = 3 and 1 ≤ p , q , r ≤ ∞ , 1 − ( 1 2 p + 1 2 q + 1 2 r ) ≥ 0 , and ∫ 0 T ∗ ‖ ‖ ∂ 3 P ( ⋅ , t ) ‖ L x 3 γ ‖ L x 1 x 2 α β + ‖ ∇ d ( ⋅ , t ) ‖ L 4 8 d t = ∞ , with 2 β + 1 γ + 2 α = k ∈ [ 2 , 3 ) and 3 k ≤ γ ≤ α < 1 k − 2 . [ABSTRACT FROM AUTHOR]

In this paper, we investigate the global regularity of Leray–Hopf weak solutions to the 3D Navier–Stokes equations in terms of pressure. In particular, we establish three Serrin-type regularity criteria in framework of the anisotropic Lebesgue spaces. Our result can be understood as a generalization of the notable works of Y. Zhou (2006) [23] , C. Cao and E. Titi (2008) [5] , and Y. Zhou and M. Pokorný (2010) [24] . [ABSTRACT FROM AUTHOR]

We study the regularity criteria for the incompressible Navier–Stokes equations based on either one item of the velocity gradient, ∂ 1 u 3 or ∂ 3 u 3 or on two items of the velocity gradient, ∂ 2 u 3 , ∂ 3 u 3 , or ∂ 1 u 3 , ∂ 2 u 3 . We improve and/or extend several results from the literature by the use of two versions of the generalized Troisi inequality. [ABSTRACT FROM AUTHOR]

We study the regularity criteria for the incompressible Navier–Stokes equations in the whole space R 3 based on one velocity component, namely u 3 , ∇ u 3 and ∇ 2 u 3 . We use a generalization of the Troisi inequality and anisotropic Lebesgue spaces and prove, for example, that the condition ∇ u 3 ∈ L β ( 0 , T ; L p ) , where 2 / β + 3 / p = 7 / 4 + 1 / ( 2 p ) and p ∈ ( 2 , ∞ ] , yields the regularity of u on ( 0 , T ] . [ABSTRACT FROM AUTHOR]

We show the following extension of the recent optimal regularity results by Kukavica & Ziane, Cao and Zhang: If u is a solution of the Navier–Stokes equations in the whole space R 3 and ∂ 3 u ∈ L p (0 , T ; L q (R 3)) , T > 0 , where 2 ∕ p + 3 ∕ q = 2 and q ∈ (3 ∕ 2 , 3 , then u is regular on (0 , T . [ABSTRACT FROM AUTHOR]