A Regularity Criterion for Semigroup Rings

An analogue of the Kunz–Frobenius criterion for the regularity of a local ring in a positive characteristic is established for general commutative semigroup rings.     

Regularity Criterion for Complete Matrix Semirings

Necessary and sufficient conditions for the regularity of complete matrix semirings over an arbitrary semiring are obtained.     

A Regularity Criterion for the 3D Incompressible Density-dependent Navier-Stokes-Allen-Cahn Equations

In this paper, we study the initial boundary problem for 3D incompressible density-dependent Navier-Stokes-Allen-Cahn equations, and give a regularity criterion for local strong solutions. Our result refines the blow-up criterion in [1].     

Regularity Criterion for a Nonhomogeneous Incompressible Ginzburg-Landau-Navier-Stokes System

Applications of Mathematics - We prove a regularity criterion for a nonhomogeneous incompressible Ginzburg-Landau-Navier-Stokes system with the Coulomb gauge in ?3. It is proved that if the...     

Regularity Criterion for Solutions of Three-Dimensional Turbulent Channel Flows

In this paper we consider the three-dimensional Navier–Stokes equations in infinite channel. We provide a regularity criterion for solutions of the three-dimensional Navier–Stokes equations in terms of the vertical component of the velocity field.     

A New Regularity Criterion for the 3D MHD Equations Involving Partial Components

In this paper, we establish a new regularity criterion for the 3D incompressible MHD equations involving partial components of the velocity gradient and magnetic fields.     

A Logarithmically Improved Regularity Criterion for the 3D Boussinesq Equations Via the Pressure

In this paper, we consider the three-dimensional Boussinesq equations, and obtain a logarithmically improved regularity criterion in terms of pressure.     

三维液晶方程一个与水平速度相关的改进正则性准则

根据速度的水平分量$u_h$的垂直导数,本文研究了3维液晶方程的整体正则性,具体地,当速度的水平分量$u_h$的垂直导数满足$\partial_3u_h\in L^p(0,T; R^3), \frac{2}{p}+\frac{3}{q}\le \frac{3}{2}$, $2\le p\le \infty$,则局部强解$(u,d)$可以光滑的延拓出$t=T$.     

Sub-Laplacians on Sub-Riemannian Manifolds

We consider different sub-Laplacians on a sub-Riemannian manifold M. Namely, we compare different natural choices for such operators, and give conditions under which they coincide. One of these operators is a sub-Laplacian we constructed previously in Gordina and Laetsch (Trans. Amer. Math. Soc., 2015). This operator is canonical with respect to the horizontal Brownian motion; we are able to define this sub-Laplacian without some a priori choice of measure. The other operator is \(\operatorname {div}^{\omega } \operatorname {grad}_{\mathcal {H}}\) for some volume form ω on M. We illustrate our results by examples of three Lie groups equipped with a sub-Riemannian structure: SU(2), the Heisenberg group and the affine group.     

A New Regularity Criterion for the Navier-Stokes Equations in Terms of the Direction of Vorticity

In this paper, we prove a new regularity criterion in terms of the direction of vorticity for the weak solution to 3-D incompressible Navier-Stokes equations.     

A New Regularity Criterion for the Navier-Stokes Equations in Terms of the Direction of Vorticity

In this paper, we prove a new regularity criterion in terms of the direction of vorticity for the weak solution to 3-D incompressible Navier-Stokes equations.     

A Regularity Result for CR Mappings Between Infinite Type Hypersurfaces

The Schwarz reflection principle in one complex variable can be stated as follows. Let M and M′ be two real analytic curves in ? and f a holomorphic function defined on one side of M, extending continuously through M, and mapping M into M′. Then f has a holomorphic extension across M. In this paper, we extend this classical theorem to higher complex dimensions for a class of hypersurfaces of infinite type.     

A New Regularity Criterion for the 3D Navier-Stokes Equations via Two Entries of the Velocity Gradient Tensor

We consider the Cauchy problem for the incompressible Navier-Stokes equations in R ³, and provide a new regularity criterion involving only two entries of the Jacobian matrix of the velocity field.     

微极流体方程组在Besov空间中弱解的正则性准则

In this paper,we investigate the regularity criterion via the pressure of weak solutions to the micropolar fluid equations in three dimensions.We obtain that for 0<α<1 if p€L^2/a(0,T;B^α∞,∞),then the weak solution(u,ω)is regular on(0,T).     

A Massera Type Criterion for Linear Functional Differential Equations with Advance and Delay

In this note, a Massera type criterion for the existence of periodic solutions for linear functional differential equations with advance and delay is established. Because of the presence of an advanced argument, the definition of the fundamental solution operator seems unknown. Hence a method different from the usual one is employed. Applications to periodic problems for nonlinear equations are also given.     

On Regularity for Beurling–Deny Type Dirichlet Forms

Characteristic examples of Beurling–Deny type Dirichlet forms are considered. The forms are identified with bilinear forms of integro-differential operators that arise as generators of jump-diffusion processes. The aim of this article is to prove Harnack inequalities for these operators and consequently Hölder regularity of weak H ¹-solutions. Moser's iteration technique is used.     

A Regularity Result of Nikolskii Type for an Evolution Equation on Compact Homogeneous Spaces

In the previous paper [2], we proposed a definition of a class of function spaces of Nikolskii type on a special class of riemannian manifolds, namely the manifolds M admitting a transitive compact connected Lie group of isometries. In this paper we continue the subject and we complete the proof that the spaces we have introduced coincide with the classical ones, defined through (finitely many) local charts of M. Moreover, our equivalent definition of these spaces allows us to prove here a regularity result for the solution of a Stefan-like problem on M, in which the initial datum is assumed to be of Nikolskii type.