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In this paper, we find a new regularity criterion of the 3D tropical climate model in terms of one component of the barotropic velocity. Our criteria of the velocity field improve the previous result by Berti et al. Moreover, we extend the regularity criteria of the Lebesgue spaces to that of larger Lorentz spaces. The key of this paper is by using interaction between partial and usual H1$$ {H}^1 $$ estimate to get upper bound of velocities. [ABSTRACT FROM AUTHOR]

We establish the regularity criterion of weak solutions to the incompressible non‐Newtonian fluids with shear thinning viscosity in the whole space R3 for 85

J. Q. Yang (2019) established a regularity criterion for the 3D shear thinning fluids in the whole space $\Bbb{R}^3$ via two velocity components. The goal of this short note is to extend this result in viewpoint of Lorentz space.

In this paper, we obtained the global existence and large time behavior of the solution for bipolar Navier-Stokes-Poisson equations under the partially smallness assumption of the initial data. Due to the complexity of bipolar Navier-Stokes-Poisson equations, we chose Green's function method instead of the classical energy method and thus discussed the regularity criterion under decaying structures in time instead of only integrability of time variable. It made the whole proof more simple and clear, meanwhile, resulted in the large time decaying estimates of the solution. It also showed the advantage of Green's function method in the study of global existence in the large perturbation framework.

Keywords: Regularity criterion; bootstrap argument; Green's function; global existence; decay rate of solution.GraphBy Jie Qi and Weike WangReported by Author; Author [Extracted from the article]

In this article, some elementary observations are is made regarding the behavior of solutions to the two-dimensional curl-free Burgers equation which suggests the distinguished role played by the scalar divergence field in determining the dynamics of the solution. These observations inspire a new divergence-based regularity condition for the two-dimensional Kuramoto–Sivashinsky equation (KSE) that provides conceptual clarity to the nature of the potential blow-up mechanism for this system. The relation of this regularity criterion to the Ladyzhenskaya–Prodi–Serrin-type criterion for the KSE is also established, thus providing the basis for the development of an alternative framework of regularity criterion for this equation based solely on the low-mode behavior of its solutions. The article concludes by applying these ideas to identify a conceptually simple modification of KSE that yields globally regular solutions, as well as providing a straightforward verification of this regularity criterion to establish global regularity of solutions to the 2D Burgers–Sivashinsky equation. The proofs are direct, elementary, and concise. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new regularity criterion in terms of the magnetic field for the local-in-time smooth solution of the two-dimensional non-resistive MHD equations. In particular, our result settles an open problem remarked by Lei et al. (2010), [11]. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigating regularity criteria for the 3D micropolar fluid equations in terms of pressure in weak Lebesgue space. More precisely, we mainly proved that the weak solution is regular on (0, T] provided that either the norm π L α , ∞ (0 , T ; L β , ∞ (R 3)) with 2 α + 3 β = 2 and 3 2 < β < ∞ or ∇ π L α , ∞ (0 , T ; L β , ∞ (R 3)) with 2 α + 3 β = 3 and 1 < β < ∞ is sufficiently small. [ABSTRACT FROM AUTHOR]

In this paper, we obtained the global existence and large time behavior of the solution for bipolar Navier-Stokes-Poisson equations under the partially smallness assumption of the initial data. Due to the complexity of bipolar Navier-Stokes-Poisson equations, we chose Green's function method instead of the classical energy method and thus discussed the regularity criterion under decaying structures in time instead of only integrability of time variable. It made the whole proof more simple and clear, meanwhile, resulted in the large time decaying estimates of the solution. It also showed the advantage of Green's function method in the study of global existence in the large perturbation framework. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the regularity criterion of weak solutions to three-dimensional magneto-micropolar fluid equations with fractional dissipation. A regularity criterion is established via the third component of the velocity fields, the micro-rotational velocity fields, and the magnetic fields.

In this paper, we prove the local well-posedness and regularity criterion of strong solutions to the 3D density-dependent incompressible Maxwell–Navier–Stokes system in a bounded and simply connected domain with vacuum. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the regularity criterion of weak solutions to three-dimensional magneto-micropolar fluid equations with fractional dissipation. A regularity criterion is established via the third component of the velocity fields, the micro-rotational velocity fields, and the magnetic fields. [ABSTRACT FROM AUTHOR]

In this paper, we show that a weak solution to 3D shear-thinning flows becomes strong provided that the gradient of one component of velocity belongs to a certain Lebesgue space class. [ABSTRACT FROM AUTHOR]

In this paper, we prove that a weak solution of the Cauchy problem for 3D unsteady flows of a generalized Newtonian fluid becomes a strong solution for 5 3 < p < 11 5 provided that the velocity field belongs to the critical space L 2 p − 1 (0 , T ; B M O (R 3)). The main key is to apply variants of Gagliardo–Nirenberg's inequality including a BMO -norm. In particular, when 2 < p < 11 5 , we derive and use a nonlinear variant of Gagliardo–Nirenberg's inequality including ‖ u ‖ B M O and ‖ | D u | p − 2 2 ∇ 2 u ‖ 2. [ABSTRACT FROM AUTHOR]

This paper is concerned with the logarithmically improved regularity criterion in terms of the middle eigenvalue of the strain tensor to the 3D Boussinesq equations in Besov spaces with negative indices. It is shown that a weak solution is regular on $ (0, T] $ provided that$ \begin{align*} \int_{0}^{T}\frac{\left\Vert \lambda _{2}^{+}(\cdot , t)\right\Vert _{\dot{B} _{\infty , \infty }^{-\delta }}^{\frac{2}{2-\delta }}}{\ln (e+\left\Vert u(\cdot , t)\right\Vert _{\dot{B}_{\infty , \infty }^{-\delta }})}dt<\infty. \end{align*} $for some $ 0<\delta <1 $. As a consequence, this result is some improvements of recent works [11,12] established by Neustupa-Penel and Miller. [ABSTRACT FROM AUTHOR]

In this article, we study the regularity criteria of the weak solutions to the Boussinesq equations involving the horizontal component of velocity or the horizontal derivatives of the two components of velocity in anisotropic Lorentz spaces. This result reveals that the velocity field plays a dominant role in regularity theory of the Boussinesq equations.

This paper is devoted to investigating regularity criteria for the 3D micropolar fluid equations in terms of pressure in weak Lebesgue space. More precisely, we prove that the weak solution is regular on $ (0, T] $ provided that either the norm $ \left\Vert \pi \right\Vert _{L^{\alpha, \infty }(0, T;L^{\beta, \infty }(\mathbb{R}^{3}))} $ with $ \frac{2}{\alpha }+ \frac{3}{\beta } = 2 $ and $ \frac{3}{2} < \beta < \infty $ or $ \left\Vert \nabla \pi \right\Vert _{L^{\alpha, \infty }(0, T;L^{\beta, \infty }(\mathbb{R} ^{3}))} $ with $ \frac{2}{\alpha }+\frac{3}{\beta } = 3 $ and $ 1 < \beta < \infty $ is sufficiently small.

The question of whether the two-dimensional inviscid Boussinesq equations can develop a finite-time singularity from general initial data is a challenging open problem. In this paper, we obtain two new regularity criteria for the local-in-time smooth solution to the two-dimensional inviscid Boussinesq equations. Similar result is also valid for the nonlocal perturbation of the two-dimensional incompressible Euler equations. [ABSTRACT FROM AUTHOR]

In this work, we investigate the regularity of a weak solution to a model for unsteady motion of an incompressible non-Newtonian fluid. Our main result is a regularity criterion for 3D flows of shear-thinning fluids via one component of velocity. [ABSTRACT FROM AUTHOR]

We consider the three-dimensional inviscid Boussinesq–Voigt system which is a regularization model for the inviscid Boussinesq equations. We prove a regularity criterion for the weak solutions (in particular, for the time-derivative of the velocity), in L p -spaces, involving first derivatives of the pressure. [ABSTRACT FROM AUTHOR]

We consider a bilinear optimal control problem associated to the following chemotaxis-consumption model in a bounded domain Ω⊂R³ during a time interval (0,T): ∂tu-Δu=-∇⋅(u∇v),∂tv-Δv=-usv+fv1Ωc, with s≥1, endowed with isolated boundary conditions and initial conditions for (u,v), u being the cell density, v the chemical concentration and f the bilinear control acting in a subdomain Ωc⊂Ω. The existence of weak solutions (u,v) to this model given f∈Lq((0,T)×Ω), for some q>5/2, has been proved in [F. Guillén-González and A. L. Corrêa Vianna Filho, Optimal Control Related to Weak Solutions of a Chemotaxis-Consumption Model, arXiv:2211.14612, 2022]. In this paper, we study a related optimal control problem in the strong solution setting. First, imposing the regularity criterion us∈Lq((0,T)×Ω) (q>5/2) for a given weak solution, we prove existence and uniqueness of global-in-time strong solutions. Then, the existence of a global optimal solution can be deduced. Finally, using a Lagrange multipliers theorem, we establish first order optimality conditions for any local optimal solution, proving existence, uniqueness and regularity of the associated Lagrange multipliers. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigating regularity criteria for the 3D micropolar fluid equations in terms of pressure in weak Lebesgue space. More precisely, we prove that the weak solution is regular on (0; T] provided that either the norm kkL;1(0;T;L1(R3)) with 2 + 3 = 2 and 32< < 1 or krkL;1(0;T;L;1(R3)) with 2 + 3 = 3 and 1 < 1 is suffciently small. [ABSTRACT FROM AUTHOR]

In this paper, we derive two regularity criteria of solutions to the nematic liquid crystal flows. More precisely, we prove that the local smooth solution (u,d)$$ \left(u,d\right) $$ is regular if and only if one of the following two conditions is satisfied: (i) ∇huh∈L2p2p−3(0,T;Lp(ℝ3)),∂3d∈L2qq−3(0,T;Lq(ℝ3)),32

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]

In this paper, we establish a regularity criterion for micropolar fluid flows in terms of the one component of the velocity in critical Morrey-Campanato space. More precisely, we show that if ... then the weak solution (u,w) is regular. [ABSTRACT FROM AUTHOR]

This work focuses on the 3D incompressible magnetohydrodynamic (MHD) equations with mixed pressure-velocity-magnetic field in view of Lorentz spaces. Our main result shows the weak solution is regular, provided that π e−|x| 2 + |u| + |b| θ ∈ L p (0, T; L q,∞), where 2 p + 3 q = 2 − θ and 0 ≤ θ ≤ 1.

In this paper, we obtain a regularity criterion via horizontal components of velocity and molecular orientations for the 3D nematic liquid crystal flows. That is the smooth solution $ (u, d) $ can be extended beyond T, provided that $ \int_{0}^{T}(||u_{h}||_{\dot{B}_{\infty, \infty}^{0}}^{2}+||\nabla d||_{\dot{B}_{\infty, \infty}^{0}}^{2}) \mbox{d} t < \infty $.

The goal of this paper is to establish a regularity criterion for the three-dimensional incompressible Hall-magnetohydrodynamic equations involving only the vorticity. Our result complements and improves some existing results. [ABSTRACT FROM AUTHOR]

In this paper, we give a BKM-type blow-up criterion, which involves only a single component of the velocity, for the inviscid axially symmetric Boussinesq system. More precisely, we will show that if the vorticity of the swirl part of the velocity belongs to L 1 (0 , T ∗ , L ∞) , then the solution is regular up to time T ∗ . At present, our results can not be easily generalized to the classical L t 1 B M O regularity criterion, which will be considered in our further works. [ABSTRACT FROM AUTHOR]

By developing the local energy bound with B ˙ ∞ , ∞ - 1 and fully using the property of B ˙ ∞ , c (N) - 1 and B ˙ p , ∞ - 1 + 3 / p ∩ B ˙ ∞ , q - 1 (p , q < ∞) , we provide a blow-up criterion to the 3D Navier–Stokes equations in the critical Besov space with - 1 negative regularity. More precisely, if u 3 ∈ L ∞ (0 , T ; B ˙ p , ∞ - 1 + 3 / p ∩ B ˙ ∞ , q - 1) and u h ∈ L ∞ (0 , T ; B ˙ ∞ , ∞ - 1) with u h (T) ∈ B ˙ ∞ , c (N) - 1 , the smooth solution u = (u h , u 3) can be extended beyond T, which improves the result of Wang-Zhang (Sci China Math 60: 637–650, 2017) and Li-Zhou (Dyn PDE 18: 71–80, 2021). [ABSTRACT FROM AUTHOR]

We investigate regularity criteria for weak solutions to the Cauchy problem of the 3d co-rotational Beris-Edwards system for nematic liquid crystals, which couples the Navier–Stokes equations for the fluid velocity u with an evolution-diffusion equations for the Q -tenser. Our results yield that for any positive constant γ > 0 , if either the negative part of the associated pressure Π satisfies Π − [ ln ⁡ (1 + Π −) ] 1 + γ ∈ L ∞ (R + ; L 3 2 , ∞ (R 3)) , or the quantity 2 Π + | u | 2 + | ∇ Q | 2 satisfies (2 Π + + | u | 2 + | ∇ Q | 2) [ ln ⁡ (1 + 2 Π + + | u | 2 + | ∇ Q | 2) ] 1 + γ ∈ L ∞ (R + ; L 3 2 , ∞ (R 3)) , then the weak solution (u , Q) , to the 3d co-rotational Beris-Edwards system, is global-in-time smooth. Here, the subscript "−" and "+" denote the negative and the nonnegative part, respectively. L 3 2 , ∞ (R 3) denotes the standard weak Lebesgue space. If Q ≡ 0 , then our results extend some previous known results from the theory of the 3d Navier–Stokes equations. [ABSTRACT FROM AUTHOR]

Consider the time-periodic viscous incompressible fluid flow past a body with non-zero velocity at infinity. This article gives sufficient conditions such that weak solutions to this problem are smooth. Since time-periodic solutions do not have finite kinetic energy in general, the well-known regularity results for weak solutions to the corresponding initial-value problem cannot be transferred directly. The established regularity criterion demands a certain integrability of the purely periodic part of the velocity field or its gradient, but it does not concern the time mean of these quantities. [ABSTRACT FROM AUTHOR]

This paper gives a further investigation on the regularity criteria for three-dimensional micropolar equations in Besov spaces. More precisely, it is proved that the weak solution (μ, ω) is regular if the velocity u satisfies. [ABSTRACT FROM AUTHOR]

In this paper, we establish a regularity criterion for the 3D nematic liquid crystal flows. More precisely, we prove that the local smooth solution $ (u, d) $ is regular provided that velocity component $ u_{3} $, vorticity component $ \omega_{3} $ and the horizontal derivative components of the orientation field $ \nabla_{h}d $ satisfy $ \begin{eqnarray*} \int_{0}^{T}|| u_{3}||_{L^{p}}^{\frac{2p}{p-3}}+||\omega_{3}||_{L^{q}}^{\frac{2q}{2q-3}}+||\nabla_{h} d||_{L^{a}}^{\frac{2a}{a-3}} \mbox{d} t

In this paper, we obtain the nth-logarithmically improved regularity criterion of smooth solutions for the incompressible Leray-\alpha-MHD model in terms of the magnetic field B. Meanwhile, the new logarithmically improved regularity criterion for the 3D Navier-Stokes equation in terms of the pressure \pi and gradient of velocity \nabla u can also be established. Especially, we explore a new logarithmically improved Serrin's criterion for the 3D Navier-Stokes equations, which improves the results of Lei and Zhou [Commn. Pure Appl. Anal. 12 (2013), pp. 2715–2719]. [ABSTRACT FROM AUTHOR]

A one-component regularity criterion for the non-resistive axially symmetric Hall-MHD system is given in this paper. More precisely, we show that strong solutions to a family of axially symmetric initial data, whose velocity and current density have trivial swirl components, could be smoothly extended beyond a possible blow-up time T ∗ if and only if the swirl component of the magnetic field h θ satisfies a Beale-Kato-Majda-type criterion. See (1.5) below. This criterion is not trivial even if the velocity field vanishes since the quantity h θ r satisfies a partial Burgers' equation in this situation. [ABSTRACT FROM AUTHOR]

Abstract This note obtains a new regularity criterion for the three-dimensional magneto-micropolar fluid flows in terms of one velocity component and the gradient field of the magnetic field. The authors prove that the weak solution ( u , ω , b ) $(u,\omega,b)$ to the magneto-micropolar fluid flows can be extended beyond time t = T $t=T$ , provided if u 3 ∈ L β ( 0 , T ; L α ( R 3 ) ) $u_{3}\in L^{\beta }(0,T;L^{\alpha }(R^{3}))$ with 2 β + 3 α ≤ 3 4 + 1 2 α , α > 10 3 $\frac{2}{\beta }+\frac{3}{\alpha }\leq \frac{3}{4}+\frac{1}{2\alpha },\alpha > \frac{10}{3}$ and ∇ b ∈ L 4 p 3 ( p − 2 ) ( 0 , T ; M ˙ p , q ( R 3 ) ) $\nabla b\in L^{\frac{4p}{3(p-2)}}(0,T;\dot{M}_{p,q}(R^{3}))$ with 1 < q ≤ p < ∞ $1< q\leq p

This paper focuses on the n-dimensional ( n = 2 , 3 ) Camassa–Holm equations with non-local diffusion of type (- Δ) s . In Gan et al. (Calc Var Partial Differ Equ. 60, 2021), they proved that with regular initial data, the finite energy weak solutions are indeed regular for all time if n / 4 < s < 1 . The purpose of this paper is to improve the result of Gan et al. Actually, we will establish the following regularity criterion: If ∇ u ∈ L q ([ 0 , T ] ; L r (R n)) with n r + 2 s q ≤ 2 s , r > n 2 s , then v is regular. As a corollary, we can obtain that if n - 2 4 < s < 1 , then the finite energy weak solutions of the fractional Camassa–Holm equations are regular. In addition, by partially appealing to some ideas used in partial regularity theory, when n = 3 , we can prove that if 1 2 < s < 1 , then the suitable weak solution of the fractional Camassa–Holm equations must be bounded. [ABSTRACT FROM AUTHOR]

In this paper, we prove some new L^p-estimates of the velocity by the technique of L^p-energy method. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we establish the regularity criterion for the weak solution of nematic liquid crystal flows in three dimensions when the L ∞ ( 0 , T ; B ˙ ∞ , ∞ − 1 ) $L^{\infty }(0,T;\dot{B}_{\infty,\infty }^{-1})$ -norm of a suitable low frequency part of ( u , ∇ d ) $(u,\nabla d)$ is bounded by a scaling invariant constant and the initial data ( u 0 , ∇ d 0 ) $(u_{0},\nabla d_{0})$ . Our result refines the corresponding one in (Liu and Zhao in J. Math. Anal. Appl. 407:557-566, 2013) and that in (Ri in Nonlinear Anal. TMA 190:111619, 2020).

In this paper, we prove a single-component BKM-type regularity criterion for the inviscid axially symmetric Hall-MHD system. More precisely, we show that if the current density is swirl-free, then the L t 1 L x ∞ boundedness of partial vorticity (corresponding to curl of the swirl part of the velocity) implies regularity of the solution. The novelty of our result is that we only impose a critical regularity criterion on one component of the velocity which is new even if we ignore the magnetic field and consider only the 3D axially symmetric Euler equations. Our results can not be easily extended to the classical L t 1 B M O norm regularity criterion at present, which will be pursued in our further works. [ABSTRACT FROM AUTHOR]

In this paper, we consider Cauchy problem of the axially symmetric Magnetohydrodynamic (MHD) system. By using energy method, we establish some regularity criteria of the solutions for the axisymmetric solutions of the three dimensional incompressible MHD system. [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].

In this paper we investigate the regularity criterion for the local-in-time smooth solution to the three-dimensional (3D) tropical climate model in the Morrey--Campanato space. It is shown that if $u $ satisfies \[\int^{T}_{0}\frac{\|\nabla u(t)\|^{\frac{2}{r}}_{\dot{\mathcal{M}}_{2,3/r}}}{\ln(\|u(t)\|_{L^2}+e)}dt