S. Gala, M. A. Ragusa
In this work, we prove a regularity criterion for micropolar fluid flows in terms of the one partial derivative of the velocity in Morrey-Campanato space.
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Regularity criteria of 3D generalized magneto-micropolar fluid system in terms of the pressure
本文受近期工作启发，研究了 Lorentz 空间中基于压力的三维广义磁微极流体系统的正则性准则。
The global well-posedness of the smooth solution to the three-dimensional (3D) incompressible micropolar equations is a difficult open problem. This paper focuses on the 3D incompressible micropolar equations with fractional dissipations $( Δ)^αu$ and $(-Δ)^βw$.Our objective is to establish the global regularity of the fractional micropolar equations with the minimal amount of dissipations. We prove that, if $α\geq \frac{5}{4}$, $β\geq 0$ and $α+β\geq\frac{7}{4}$, the fractional 3D micropolar equations always possess a unique global classical solution for any sufficiently smooth data. In addit…
In the analysis of PDEs, regularity of often measured in terms of Sobolev, H{ö}lder, Besov or Lipschitz spaces, etc. However, sometimes a gain of regularity can also be expressed just in terms of Lebesgue spaces, by passing from a singular setting to a less singular one. In this article we will obtain a gain of integrability for weak solutions of the micropolar fluid equations using as general framework Morrey spaces, which is a very useful language to study regularity in PDEs. An interesting point is that the two variables of the micropolar fluid equations can be studied separately.
In this article, we study regularity criteria for the 3D micropolar fluid equations in terms of one partial derivative of the velocity. It is proved that if \begin{equation*} \int^{T}_{0}\|\partial_{3}u\|^{\frac{2}{1-r}}_{\dot{B}^{-r}_{\infty,\infty}} dt<\infty \quad \text{with} \quad 0< r<1, \end{equation*} then, the solutions of the micropolar fluid equations actually are smooth on $(0, T)$. This improves and extends many previous results.
Global regularity for the 2D micropolar Rayleigh-Bénard convection system with velocity zero dissipation and temperature critical dissipation
针对速度零耗散、微旋转速度 Laplace 耗散和温度临界耗散的二维微极 Rayleigh-Bénard 系统，论文借助组合量与 Littlewood-Paley 分解证明其解的全局正则性。
The incompressible Micropolar system is given by two coupled equations: the first equation gives the evolution of the velocity field u while the second equation gives the evolution of the microrotation field $ω$. In this article we will consider regularity problems for weak solutions of this system. For this we will introduce the new notion of partial suitable solutions, which imposes a specific behavior for the velocity field u only, and under some classical hypotheses over the pressure, we will obtain a h{ö}lderian gain for both variables u and $ω$.
Existence theorem and blow-up criterion of the strong solutions to the Magneto-micropolar fluid equations
本文研究了$\R^3$中的磁微极流体方程，证明了$s> {3/2}$在$H^s(\R^3)$中具有初值的强解的存在性，并建立了其爆破准则.我们主要利用Littlewood-Paley分解，得到了光滑解$(u,ω,b)$的Beale-Kato-Majda型爆破准则，它只依赖于速度的涡度$\nabla\times u$.
We develop here a particular version of the partial regularity theory for the Magneto-Micropolar equations (MMP) where a perturbation term is added. These equations are used in some special cases, such as in the study of the evolution of liquid cristals or polymers, where the classical Navier-Stokes equations are not an accurate enough model. The incompressible Magneto-Micropolar system is composed of three coupled equations: the first one is based in the Navier-Stokes system, the second one considers mainly the magnetic field while the last equation introduces the microrotation field represen…
We consider the rigorously derived thin shell membrane $Γ$-limit of a three-dimensional isotropic geometrically nonlinear Cosserat micropolar model and deduce full interior regularity of both the midsurface deformation $m:ω\subset{\mathbb R}^2\to{\mathbb R}^3$ and the orthogonal microrotation tensor field $R:ω\subset{\mathbb R}^2\to SO(3)$. The only further structural assumption is that the curvature energy depends solely on the uni-constant isotropic Dirichlet type energy term $|DR|^2$. We use Rivière's regularity techniques of harmonic map type systems for our system which couples harmonic m…
In this paper, we consider weak solutions of the Euler-Lagrange equation to a variational energy functional modeling the geometrically nonlinear Cosserat micropolar elasticity of continua in dimension three, which is a system coupling between the Poisson equation and the equation of $p$-harmonic maps ($2\le p\le 3$). We show that if a weak solutions is stationary, then its singular set is discrete for $2<p<3$ and has zero $1$-dimensional Hausdorff measure for $p=2$. If, in addition, it is a stable-stationary weak solution, then it is regular everywhere when $p\in [2, \frac{32}{15}]$.
Some remarks about the stationary Micropolar fluid equations: existence, regularity and uniqueness
本文研究定常微极流体方程，推导速度场与微旋转场的耦合方程，探讨弱解存在性、正则性及唯一性，证明仅速度场满足无穷远衰减条件时平凡解唯一。
Partial regularity and $L^3$-norm concentration effects around possible blow-up points for the micropolar fluid equations
本文研究微极流体方程，在速度场附加条件下证明弱解的光滑性，并给出速度场$L^3_x$范数在可能奇异时间附近的集中效应。
On the regularity criteria of weak solutions to the micropolar fluid equations in Lorentz space
本文在Lorentz空间$L^{p,\infty}(\mathbb{R}^3)$中研究了三维微极流体方程的弱解的正则性和光滑解的爆破准则.证明了若$u\in L^q(0,T;L^{p,\infty}(\mathbb{R}^3))$对于$\frac2q+\frac3p\le 1$和$3<p\le \infty$ ;或$\nabla u\in L^q(0,T;L^{p,\infty}(\mathbb{R}^3))$对于$\frac2q+\frac3p\le 2$和$\frac32<p\le \infty$ ;或压力$P\in L^q(0,T;L^{p,\infty}(\mathbb{R}^3))$对于$\frac2q+\frac3p\le 2$和$\frac32<p\le \infty$ ;或$\nabla P\in L^q(0,T;L^{p,\infty}(\mathbb{R}^3))$对于$\frac2q+\frac3p\le 3$和$1<p\le \infty$，则满足能量不等式的弱解$(u,ω)$是$[0,T)$上的光滑解.
On the initial- and boundary-value problem for 2D micropolar equations with only angular velocity dissipation
本文研究了在光滑有界区域上仅具角速度耗散的二维微极方程的初边值问题边值问题。这里的目的是建立整体的存在性和唯一性的解决方案，通过施加自然边界条件和最小的正则性假设的初始数据。此外，当初始数据较规则时，整体解具有较高的规则性。为了获得这些结果，我们克服了两个主要的困难，一个是由于缺乏充分的耗散，一个是由于边界条件。除了全局正则性问题，我们还研究了大时间行为的解决方案，并获得明确的衰减率。
Micropolar fluids with initial angular velocities in non-homogeneous Sobolev spaces of order $-1/2$
本文针对微极流体提出分数阶能量方法，在初始角速度具有-1/2阶负Sobolev正则性的非齐次空间中开展研究，并指出该框架可推广至带粘性的Boussinesq系统，为Navier-Stokes方程耦合变量的奇异性分析提供新工具。
Stability of vertically charged steady magnetic field in 3D incompressible magneto-micropolar fluids without magnetic and angular viscosity in a strip domain
研究三维不可压缩磁微极方程在带状区域无磁和角粘性时的正则性与稳定性问题，给出稳态设置，证明小扰动有全局正则稳定解且解随时间趋于无穷以近指数速率收敛到稳态。
Global dynamics of Kato's solutions for the 3D incompressible micropolar system
本文研究临界Besov空间中三维不可压微极方程解的全局动力学，通过谱分析发现低频阻尼效应，结合Gevrey方法与扩展Coifman-Meyer理论，在更一般初值正则性下建立全局适定性并推导高阶导数衰减。
Local and global strong solutions to a reduced model for inviscid micropolar fluids
研究无粘微极性流体二维简化模型的适定性，在Besov空间中建立局部强解的存在唯一性，并在次临界条件下证明全局存在性。
Global Yudovich-type solutions to a reduced model for micropolar fluids with zero viscosity
本文研究了二维微极流体简化模型在低正则性下的适定性，证明了全局唯一性Yudovich型解，扩展了Yudovich框架到扰动不可压欧拉方程的系统。
Regularity issues for Cosserat continua and $p$-harmonic maps
对于极小的几何非线性Cosserat模型的微极弹性的连续体，我们证明内部Hölder正则性，孤立奇点，可能是可能的，如果从模型的指数$p$是$2$或在$(\frac{32}{15},3)$。完全连续的障碍原来是存在某些极小化齐次$p$ -调和映射到$S^3$。对于这些，我们稍微改进现有的正则性定理，以实现我们的结果的Cosserat模型。
$L^p$-regularity of a geometrically nonlinear system in supercritical dimensions
本文研究了源自几何非线性平坦Cosserat微极模型的系统，在超临界维度下建立了内部正则性和精确的$L^p$正则性，扩展了Gastel和Neff在临界维度下的工作。
The combined limits of large bulk viscosity and small angular viscosity for the 2D compressible micropolar fluid equations
本文研究二维可压缩微极流体方程大体黏度与小角黏度组合极限，证明解收敛到不可压缩系统并给出收敛速率，且无需额外技术假设。
A regularity criterion for the dissipative quasi-geostrophic equations
在混合时空Besov空间中建立了耗散拟地转方程弱解的正则性准则。
Boundary characteristic point regularity for semilinear reaction-diffusion equations: Towards an ODE criterion
研究了一类半线性热方程的边界特征点正则性，得到了其正则性的一个ODE准则。推广到高阶半线性抛物问题进行了讨论。
A regularity criterion for the 3D full compressible magnetohydrodynamic equations with zero heat conductivity
建立了有界区域内真空和零热导率条件下三维全可压缩磁流体动力学方程组的正则性准则。
A Refinement of the Local Serrin--Type Regularity A refinement of the local Serrin--type regularity criterion for a suitable weak solution to the Navier--Stokes equations
我们给出了Navier-斯托克斯方程在时空点（x_0，t_0）上弱解v的正则性的一个新判据.该判据仅在（x_0，t_0）的后向邻域内，在顶点位于（x_0，t_0）点的时空抛物面的外部，对v施加了一个Serrin型可积条件。我们对抛物面内部的解不作特殊的假设。
In this paper we study the regularity problem of a three dimensional chemotaxis-Navier-Stokes system on a periodic domain. A new regularity criterion in terms of only low modes of the oxygen concentration and the fluid velocity is obtained via a wavenumber splitting approach. The result improves many existing criteria in the literature.
In this paper, we study regularity of weak solutions to the incompressible Boussinesq equations in $\mathbb{R}^{3}\times (0,T)$. The main goal is to establish the regularity criterion in terms of one velocity component and the gradient of temperature in Lorentz spaces.
A Weighted Regularity Criterion for Suitable Weak Solutions of Incompressible Non-Newtonian Fluids
本文提出了一种基于Caffarelli--Kohn--Nirenberg不等式的加权正则性准则，用于研究不可压缩非牛顿流体的合适弱解。
A new local regularity criterion for suitable weak solutions of the Navier--Stokes equations in terms of the velocity gradient
研究了三维不可压Navier-斯托克斯方程适当弱解的部分正则性。人们曾多次尝试改进Caffarelli-Kohn-Nirenberg准则（1982）。我们用直接法给出了CKN准则的一个改进版本，它也提供了Seregin准则（2007）中的定量关系。
A Serrin-type regularity criterion for the Navier-Stokes equations via one velocity component
研究了三维Navier-斯托克斯方程的Cauchy问题，证明了一些只涉及一个速度分量的标度不变正则性准则.
Boundary Characteristic Point Regularity for Navier-Stokes Equations: Blow-up Scaling and Petrovskii-type Criterion (a Formal Approach)
本文证明了抛物面边界上Dirichlet条件为零的三维Navier-斯托克斯方程的倒抛物面顶点的Wiener正则性是由热方程的Petrovskii判据（1934）给出的，即，非线性对流项不影响正则性结果。
On regularity of a boundary point for higher-order parabolic equations: towards Petrovskii-type criterion by blow-up approach
证明了对于双调和方程，典型抛物面顶点的最优正则性准则可以用1934年导出的热方程的Petrovskii型Osgood-Dini积分条件表示.讨论了第一Fourier系数法在其它偏微分方程中的推广。综述了椭圆和抛物型偏微分方程的边界点正则性。
An anisotropic partial regularity criterion for the Navier-Stokes equations
本文研究不可压Navier-斯托克斯方程适当弱解的部分正则性。我们证明了一个内部正则性准则只涉及一个组件的速度。也就是说，如果$(u,p)$是一个合适的弱解，并且在时空柱$Q_r(x_0,t_0)$上只涉及$u_3$的某个标度不变量很小，那么$u$在$(x_0,t_0)$上是正则的。
In my previous paper I have contrived a Ginzburg-Landau heat flow with a time-dependent parameter and by using it, I constructed a harmonic heat flow into spheres with a monotonical inequality and a reverse Poincaré inequality. This paper establishes these two energy inequalities near the boundary and then by making the best of them, we discuss a partial boundary regularity. In addition to it, we demonstrate a whole domain's regularity under "the one-sided condition." This has been proposed by S.Hildebrandt and K.-O.Widman.
Perron's method and Wiener's criterion have entirely solved the Dirichlet problem for the Laplace equation. Since then, this approach has attracted the attention of many mathematicians for applying these ideas in the more general equations. So, in this paper, we extend the Perron method and the Wiener criterion to the $G(\cdot)$-Laplace equation.
Regularity criterion for 3D Navier-Stokes Equations in Besov spaces
得到了三维Navier-斯托克斯方程的Leray-Hopf弱解$u$的几个正则性准则.结果表明，若速度分量$\nabla_{h}{u}$（或$ \nabla{u_3}$）的梯度满足$L^{q}(0,T; \dot{B}_{p,r}^{s}(\mathbb{R}^{3}))$类中的附加条件，则弱解$u$成为正则解，其中$\nabla_{h}=(\partial_{x_{1}},\partial_{x_{2}})$为水平梯度算子.此外，我们还考虑了Navier-斯托克斯方程在$\mathbb{R}^3$中弱解的各向异性正则性准则。最后，在给出$\partial_3u_3$的充分条件时，我们还得到了进一步的正则性准则.
In this paper, we will prove a regularity criterion that guarantees solutions of the Navier--Stokes equation must remain smooth so long as the the vorticity restricted to a plane remains bounded in the scale critical space $L^4_t L^2_x$, where the plane may vary in space and time as long as the gradient of the vector orthogonal to the plane remains bounded. This extends previous work by Chae and Choe that guaranteed that solutions of the Navier--Stokes equation must remain smooth as long as the vorticity restricted to a fixed plane remains bounded in family of scale critical mixed Lebesgue spa…
On the interior regularity criterion and the number of singular points to the Navier-Stokes equations
建立了三维Navier-斯托克斯方程弱解的内部正则性准则，使得在局部标度不变范数下，速度的垂直部分可以很大.作为应用，我们改进了Ladyzhenskaya-Prodi-Serrin准则和Escauriza-Seregin-Šverák准则。我们还证明了如果弱解$u$对某些$3<p<\infty$满足$$ \|u(\cdot,t)\|_{L^p}\leq C(-t)^{\frac {3-p}{2p}} $$，则奇点的数目是有限的。
Regularity bounds for a Gevrey criterion in a kernel-based regularization of the Cauchy problem of elliptic equations
本文给出了用正则化方法求解椭圆型方程柯西问题时Gevrey准则的正则性界。当使用正则化时，人们知道检查这样的标准基本上是有问题的，尽管它对工程环境很重要。因此，解决这个障碍有助于我们改进一些正则化方法在现实世界中的应用。这项工作也考虑了幂律非线性的存在。
In this paper we present two results: (1) A data assimilation algorithm for the 3D Navier-Stokes equation (3D NSE) using nodal data, and, as a consequence (2) a novel regularity criterion for the 3D NSE based on finitely many observations of the velocity. The data assimilation algorithm we employ utilizes nudging, a method based on a Newtonian relaxation scheme motivated by feedback-control. The observations, which may be either modal, nodal or volume elements, are drawn from a weak solution of the 3D NSE and are collected almost everywhere in time over a finite grid and our results, including…
An $\eps$-regularity criterion at one scale for 3-D Navier-Stokes equations and improved upper bound on the box dimension of the interior singular set
本文针对三维Navier-Stokes方程，通过均值波动分解的由粗到精估计得到单尺度ε正则性准则，进而将内部奇异集的盒维数上界改进为15/13。
Regularity criterion and energy conservation for the supercritical Quasi-Geostrophic equation
本文研究二维超临界准地转（SQG）方程的正则性和能量守恒问题。我们应用分裂耗散波数的方法得到一个新的正则性条件，它比所有的Prodi-Serrin型正则性条件弱。证明了超临界SQG在$L^2(0,T; B^{1/2}_{2,c(\mathbb N)})$中的任何弱解都满足能量等式。
Global regularity criterion for the 3D Navier-Stokes equations with large data
本文研究了三维不可压Navier-Stokes方程斯托克斯方程Cauchy问题强解的整体正则性。证明了具有$2\leqα\leq4$的$\nabla u\in L^α\left( 0,T,L^{2}\left( Ω\right) \right) $的强解的全局存在性.
Remarks on Interior Regularity Criterion for an Axially Symmetric Suitable Weak Solution to the Navier Stokes Equations
我们证明了，如果v是Navier-Stokes方程的轴对称适当弱解（在L。卡法雷利河科恩公司Nirenberg）使得v的径向分量具有更高的正则性（即满足加权Serrin-Prodi型条件），则v的所有分量都是正则的。
A regularity criterion in multiplier spaces to Navier-Stokes equations via the gradient of one velocity component
本文在乘子空间中利用单个速度分量的梯度，建立了三维不可压缩Navier-Stokes方程弱解的正则性准则。
Regularity criterion for the 3D Hall-magneto-hydrodynamics
本文研究了三维不可压缩电阻粘性霍尔磁流体动力学（Hall-MHD）系统的正则性问题。湍流的Kolmogorov 41唯象理论预测存在临界波数，在该临界波数以上，高频部分由流体方程中的耗散项支配。受这一想法的启发，我们应用波数分裂与能通量估计相结合的方法来获得新的正则性准则。这里提出的正则性条件是弱于现有的条件（Prodi-Serrin型标准）的三维霍尔MHD系统。
In this paper we deal with the 3D tropical climate model with damping terms in the equation of the barotropic mode $u$ and in the equation of the first baroclinic mode $v$ of the velocity, and we establish a regularity criterion for this system thanks to which the local smooth solution $(u, v, θ)$ can actually be extended globally in time.
A regularity criterion for the Navier-Stokes equations via two entries of the velocity Hessian tensor
考虑了不可压Navier-斯托克斯方程在$\R^3$中的Cauchy问题，给出了保证解光滑的充分条件.它只涉及速度Hessian的两个项。
Stress regularity in quasi-static perfect plasticity with a pressure dependent yield criterion
本文研究了平面各向同性线弹性-理想塑性材料在Drucker-Prager或Mohr-Coulomb屈服条件下的应力张量的正则性。在适当的数据假设下，证明了应力张量具有局部平方可积的空间梯度。作为推论，通常的测量理论流动法则使用应力的准连续表示以强形式表示。
In this paper, we continue our work in [15] to derive ε-regularity criteria at one scale without pressure for suitable weak solutions to the Navier-Stokes equations. We establish a $\varepsilon$-regularity criterion below of suitable weak solutions, for any $δ>0$, $$\iint_{Q(1)}|u|^{\frac{5}{2}+δ}dxdt\leq \varepsilon.$$ As an application, we extend the previous corresponding results concerning the improvement of the classical Caffarelli--Kohn--Nirenberg theorem by a logarithmic factor.
A smallness regularity criterion for the 3D Navier-Stokes equations in the largest class
本文考虑三维Navier-斯托克斯方程，证明了如果速度场的$\dot B^{-1}_{\infty,\infty}$ -范数足够小，则解实际上是经典的.
Regularity Criterion for the Three-dimensional Boussinesq Equations
我们证明了三维Boussinesq方程的解在时刻T不爆破，如果$\| u_{\le Q}\|_{B^1_{\infty, \infty}}$在$(0, T)$上可积，其中$u_{\le Q }$表示速度$u$的Littlewood-Paley投影的低模.
A uniqueness and regularity criterion for Q-tensor models with Neumann boundary conditions
在张量$Q$的齐次Neumann边界条件下，给出了模拟液晶的$Q$ -张量系统的正则性判据.从仅对速度场${\bf u}$施加的一个判据出发，证明了两个结果：弱解的唯一性和时间导数$(\partial_t {\bf u},\partial_t Q)$在时间上的整体弱正则性.本文推广了[F. Guillén-González，M.A. Rodríguez-Bellido \& M.A. Rojas-Medar，三维液晶模型正则性和唯一性的充分条件，Math. Nachr. 282（2009），no. 6，846-867]，其中$({\bf u},{\bf d})$表示液晶分子的取向矢量。
Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor
在本文中，我们提供了一个充分条件，仅用梯度张量的九个元素中的一个，即，利用速度向量场的雅可比矩阵，研究了三维Navier-斯托克斯方程强解在全空间的整体正则性，以及周期边界条件的情形.
The paper deals with the regularity criterion for the weak solutions to the 3D Boussinesq equations in terms of the partial derivatives in Besov spaces. It is proved that the weak solution $(u,θ)$ becomes regular provided that $(\nabla_{h}u,\nabla_{h}θ)\in L^{\frac{8}{3}}(0,T;\dot{B}_{\infty ,\infty}^{-1}(\mathbb{R}^{3}))$ Our results improve and extend the well-known results by Fang-Qian for the Navier-Stokes equations.
In this paper, we consider the local regularity of suitable weak solutions to the 3D incompressible Navier-Stokes equations. By means of the local pressure projection introduced by Wolf in [15,16], we present a $\varepsilon$-regularity criterion below of suitable weak solutions $$ \iint_{Q(1)}|u|^{20/7}dxdt\leq \varepsilon, $$ which gives an improvement of previous corresponding results obtained in Chae and Wolf [3, Arch. Ration. Mech. Anal., 225: 549-572, 2017], in Guevara and Phuc [6, Calc. Var., 56:68, 2017] and in Wolf [16, Ann. Univ. Ferrara, 61: 149-171, 2015].
The regularity criterion for 3D Navier-Stokes Equations
本文基于梯度张量的九个元素之一，建立了Navier-斯托克斯方程解正则的充分条件。我们改进了C.S. Cao，E.S. Titi（Arch. Rational Mech.Anal. 202（2011）919-932）和Y. Zhou，M. Pokorn $\acute{y}$（非线性23，1097-1107（2010））。
A single-component regularity criterion and Inviscid limit of axially symmetric MHD-Boussinesq system
该研究针对无粘轴对称MHD-Boussinesq系统给出仅含速度水平旋度分量的临界BKM型爆破准则，同时分析粘性系统的无粘极限并得到粘性系数趋于零时的收敛速率。
A Liouville theorem for the planer Navier-Stokes equations with the no-slip boundary condition and its application to a geometric regularity criterion
本文建立了半平面中Navier-斯托克斯方程在无滑移边界条件下向后整体解的Liouville型结果。没有对涡量和速度场的空间衰减作任何假设。我们用涡量方程代替原来的Navier-斯托克斯方程。作为应用，我们推广了三维半空间中Navier-斯托克斯方程在无滑移边界条件下的几何正则性准则。
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