ON THE REGULARITY CRITERIA OF THE 3D NAVIER-STOKES EQUATIONS IN CRITICAL SPACES

Regularity criteria of Leray-Hopf weak solutions to the three-dimensional Navier-Stokes equations in some critical spaces such as Lorentz space, Morrey space and multiplier space are derived in terms of two partial derivatives, θ1u1, θ2u2, of velocity fields.

REGULARITY CRITERIA FOR WEAK SOLUTION TO THE 3D MAGNETOHYDRODYNAMIC EQUATIONS

In this article, regularity criteria for the 3D magnetohydrodynamic equations are investigated. Some sufficient integrability conditions on two components or the gradient of two components of u ＋ B and u - B in Morrey-Campanato spaces are obtained.

THE REGULARITY CRITERIA OF WEAK SOLUTIONS TO 3D AXISYMMETRIC INCOMPRESSIBLE BOUSSINESQ EQUATIONS

In this paper,we obtain new regularity criteria for the weak solutions to the three dimensional axisymmetric incompressible Boussinesq equations.To be more precise,under some conditions on the swirling component of vorticity,we can conclude that the weak solutions are regular.

LOCAL REGULARITY CRITERIA OF A SUITABLE WEAK SOLUTION TO MHD EQUATIONS

We present a regularity condition of a suitable weak solution to the MHD equations in three dimensional space with slip boundary conditions for a velocity and magnetic vector fields. More precisely, we prove a suitable weak solution are HSlder continuous near boundary provided that the scaled mixed Lx,t^p,q-norm of the velocity vector field with 3/p ＋ 2/q 〈 2, 2 〈 q 〈 ∞ is sufficiently small near the boundary. Also, we will investigate that for this 3 2〈3 solution U ∈ Lx,t^p,q with 1 〈 3＋p ＋2/q＋≤3/2, 3 〈 p 〈 ∞, the Hausdorff dimension of its singular set is no greater than max{p, q}（3/q＋2/q- 1）.

Remarks on the Regularity Criteria of Weak Solutions to the Three-dimensional Micropolar Fluid Equations

This paper is investigate the regularity criteria of weak solutions to the three-dimensional microp- olar fluid equations. Several sufficient conditions in terms of some partial derivatives of the velocity or the pressure are obtained.

On Mixed Pressure-Velocity Regularity Criteria to the Navier-Stokes Equations in Lorentz Spaces

In this paper the authors derive regular criteria in Lorentz spaces for LerayHopf weak solutions v of the three-dimensional Navier-Stokes equations based on the formal equivalence relationπ≌|v|^(2),whereπdenotes the fluid pressure and v denotes the fluid velocity.It is called the mixed pressure-velocity problem(the P-V problem for short).It is shown that if(π/(e-^|(x)|^(2)+|v|^(θ)∈L^(p)(0,T;L^(q,∞)),where 0≤θ≤1 and 2/p+3/q=2-θ,then v is regular on(0,T].Note that,ifΩ,is periodic,e^(-|x|)^(2) may be replaced by a positive constant.This result improves a 2018 statement obtained by one of the authors.Furthermore,as an integral part of the contribution,the authors give an overview on the known results on the P-V problem,and also on two main techniques used by many authors to establish sufficient conditions for regularity of the so-called Ladyzhenskaya-Prodi-Serrin(L-P-S for short)type.

Remarks on the Regularity Criteria of Three-Dimensional Navier-Stokes Equations in Margin Case

In the study of the regularity criteria for Leray weak solutions to threedimensional Navier-Stokes equations, two sufficient conditions such that the horizontal velocity u satisfies u∈L2（0,T;BMO（R3）） or u∈L^2/1＋r（0,T;B∞,∞（R3）） for 0 〈 r 〈 1 are considered.

On Mixed Pressure-Velocity Regularity Criteria to the Navier-stokes Equations in Lorentz Spaces,Part Ⅱ:The Non-slip Boundary Value Problem

This paper is a continuation of the authors recent work[Beirao da Veiga,H.and Yang,J.,On mixed pressure-velocity regularity criteria to the Navier-Stokes equations in Lorentz spaces,Chin.Ann.Math.,42(1),2021,1-16],in which mixed pressure-velocity criteria in Lorentz spaces for Leray-Hopf weak solutions of the three-dimensional Navier-Stokes equations,in the whole space R^(3) and in the periodic torus T^(3),are established.The purpose of the present work is to extend the result of mentioned above to smooth,bounded domains Ω,under the non-slip boundary condition.Let π denote the fluid pressure and v the fluid velocity.It is shown that if π/(1+|v|^(θ))∈L^(p)(0,T;L^(q,∞)(Ω)),where 0≤θ≤1,and 2/p+3/q=2-θwith p≥2,then v is regular on Ω×(0,T].

Regularity Criteria of Axisymmetric Weak Solutions to the 3D Magnetohydrodynamic Equations

In this paper, we study the regularity criteria for axisymmetric weak solutions to the MHD equa- tions in R3. Let we, Jo and ue be the azimuthal component of w, J and u in the cylindrical coordinates, respectively. Then the axisymmetric weak solution （u,b） is regular on （0, T） if （wo,Jo） E Lq（O,T;Lp） or （oae,V（uoeo）） e Lq（0,T;Lp） with 3 ＋ 2 〈 2, 3 〈 p 〈 oo. In the endpoint case, one needs conditions （we, Jo） C LI（0, T;B∞∞） or （wo, V（uoeo）） C LI（0, T;B ∞∞）.

REGULARITY OF WEAK SOLUTIONS TO MAGNETO-MICROPOLAR FLUID EQUATIONS

In this article, we study the regularity of weak solutions and the blow-up criteria for smooth solutions to the magneto-micropolar fluid equations in R3. We obtain the classical blow-up criteria for smooth solutions （u,w, b）, i.e., u ∈ Lq（0, T; LP（R3） for 2/q＋3/P≤ 1with 3〈P≤∞,u∈C（[0,T）;L3（R3））or△u∈Lq（0,T,LP）for 3/2〈P≤∞ satisfying 2/q＋3/P≤ 2. Moreover, our results indicate that the regularity of weak solutions is dominated by the velocity u of the fluid. In the end-point case p = ∞, the blow-up criteriacan be extended to more general spaces △u E L1 （0, T; B0∞,∞（R3）.

Regularity Criterion of Weak Solutions to the 3D Micropolar Fluid Equations in Besov Space

This paper studies the regularity criterion of weak solutions to the micropolarfluid equations in three dimensions. Let (■u∈L2/2-γ(0,T;B^-γ∞,∞),■ω∈L^2(0,T;B^-1∞,∞)),it is showed that the weak solution(u,ω) is globally regular for the case 0 < γ < 2.

GLOBAL REGULARITY OF 2D INCOMPRESSIBLE MAGNETO-MICROPOLAR FLUID EQUATIONS WITH PARTIAL VISCOSITY

This paper studies the global regularity of 2D incompressible anisotropic magnetomicropolar fluid equations with partial viscosity. Ma [22](Ma L. Nonlinear Anal: Real World Appl, 2018, 40: 95–129) examined the global regularity of the 2D incompressible magnetomicropolar fluid system for 21 anisotropic partial viscosity cases. He proved the global existence of a classical solution for some cases and established the conditional global regularity for some other cases. In this paper, we also investigate the global regularity of 12 cases in [22]and some other new partial viscosity cases. The global regularity is established by providing new regular conditions. Our work improves some results in [22] in this sense of weaker regular criteria.

Conditional Regularity of Weak Solutions to the 3D Magnetic Bénard Fluid System

This paper concerns about the regularity conditions of weak solutions to the magnetic Benard fluid system in R^(3).We show that a weak solution(u,b,θ)(·,t)of the 3D magnetic Benard fluid system defined in[0,T),which satisfies some regularity requirement as(u,b,θ),is regular in R^(3)x(0,T)and can be extended as a C∞solution beyond T.

Liquid Crystal Flows with Regularity in One Direction

In this paper, we consider the Cauchy problem for the model of liquid crystal. We show that if the velocity field u satisfies 3u∈Lp（0,T;Lq（／mathbb{R}3））, ／frac{2}{p}＋／frac{3}{q}＋=1＋／frac{1}{q}, 2〈q ≤ ∞, then the solution is in fact smooth.

Regularized multiple criteria linear programs for classification

Although multiple criteria mathematical program （MCMP）, as an alternative method of classification, has been used in various real-life data mining problems, its mathematical structure of solvability is still challengeable. This paper proposes a regularized multiple criteria linear program （RMCLP） for two classes of classification problems. It first adds some regularization terms in the objective function of the known multiple criteria linear program （MCLP） model for possible existence of solution. Then the paper describes the mathematical framework of the solvability. Finally, a series of experimental tests are conducted to illustrate the performance of the proposed RMCLP with the existing methods： MCLP, multiple criteria quadratic program （MCQP）, and support vector machine （SVM）. The results of four publicly available datasets and a real-life credit dataset all show that RMCLP is a competitive method in classification. Furthermore, this paper explores an ordinal RMCLP （ORMCLP） model for ordinal multigroup problems. Comparing ORMCLP with traditional methods such as One-Against-One, One-Against-The rest on large-scale credit card dataset, experimental results show that both ORMCLP and RMCLP perform well.

Markov branching processes with immigration-migration and resurrection

We consider a modified Markov branching process incorporating with both state-independent immigration-migration and resurrection. The effect of state-independent immigration-migration is firstly in- vestigated in detail. The explicit expressions for the extinction probabilities and mean extinction times are presented. The ergodicity and stability properties of the process incorporating with resurrection structure are then investigated. The conditions for recurrence, ergodicity and exponential ergodicity are obtained. An explicit expression for the equilibrium distribution is also presented. As a preparation, the criteria for regularity and uniqueness for such structure are firstly established.