Regularity Criteria for the Three-dimensional MHD Equations

In this paper, we consider regularity criteria for solutions to the 3D MHD equations with incompressible conditions. By using some classical inequalities, we obtain the regularity of strong solutions to the three-dimensional MHD equations under certain sufficient conditions in terms of one component of the velocity field and the magnetic field respectively.

ON VORTEX ALIGNMENT AND THE BOUNDEDNESS OF THE Lq-NORM OF VORTICITY IN INCOMPRESSIBLE VISCOUS FLUIDS

We show that the spatial L q-norm(q>5/3)of the vorticity of an incompressible viscous fluid in R^3 remains bounded uniformly in time,provided that the direction of vorticity is Hölder continuous in space,and that the space-time L q-norm of vorticity is finite.The Hölder index depends only on q.This serves as a variant of the classical result by Constantin-Fefferman(Direction of vorticity and the problem of global regularity for the Navier-Stokes equations,Indiana Univ.J.Math.42(1993),775-789),and the related work by Grujić-Ruzmaikina(Interpolation between algebraic and geometric conditions for smoothness of the vorticity in the 3D NSE,Indiana Univ.J.Math.53(2004),1073-1080).

ON CONTINUATION CRITERIA FOR THE FULL COMPRESSIBLE NAVIER-STOKES EQUATIONS IN LORENTZ SPACES

In this paper,we derive several new sufficient conditions of the non-breakdown of strong solutions for both the 3D heat-conducting compressible Navier-Stokes system and nonhomogeneous incompressible Navier-Stokes equations.First,it is shown that there exists a positive constantεsuch that the solution(ρ,u,θ)to the full compressible Navier-Stokes equations can be extended beyond t=T provided that one of the following two conditions holds:(1)ρ∈L^(∞)(0,T;L^(∞)(R^(3)),u∈L^(p,∞)(0,T;L^(q,∞)(R^(3)))and ||u||_(L^(p,∞)(0,T;L^(q,∞)(R^(3))))≤ε,with 2/p+3/q=1,q>3;(0.1)(2)λ<3μ,ρ∈L^(∞)(0,T;L^(∞)(R^(3)),θ∈L^(p,∞)(0,T;L^(q,∞)(R^(3)))and ||θ||_(L^(p,∞)(0,T;L^(q,∞)(R^(3))))≤ε,with 2/p+3/q=1,q>3/2(0.2)To the best of our knowledge,this is the first continuation theorem allowing the time direction to be in Lorentz spaces for the compressible fluid.Second,we establish some blow-up criteria in anisotropic Lebesgue spaces for the finite blow-up time T^(*):(1)assuming that the pair(p,q) satisfies 2/p+1/q_(1)+1/q_(2)+1/q_(3)=1(1<q_(i)<∞)and (1.17),then lim sup_((t→T*))||ρ||_(L^(∞)(0,t;L^(∞)(R^(3))))+||u||_(Lp(0,t;L_(1)^(q_(1))L_(2)^(q_(2))L_(3)^(q_(3))(R^(3)))))=∞;(0.3)(2)letting the pair(p,q)satisfy 2/p+1/q_(1)+1/q_(2)+1/q_(3)=1(1<q_(i)<∞)and(1.17),then lim sup (t→T*)||ρ||_(L^(∞)(0,t;L^(∞)(R^(3))))+||θ||_(Lp(0,t;L_(1)^(q_(1))L_(2)^(q_(2))L_(3)^(q_(3))(R^(3)))))=∞=∞,(λ<3μ).(0.4)Third,without the condition on p in(0.1)and(0.3),the results also hold for the 3 D nonhomogeneous incompressible Navier-Stokes equations.The appearance of a vacuum in these systems could be allowed.

一种时间分数阶扩散方程的初边值问题

讨论了一种时间分数阶扩散方程的初边值问题。首先利用分离变量法求出了方程的解,然后借助于拉普拉斯变换讨论了方程的强解、连续性和正则性等性质。

关于二相障碍问题几点注记

在Orlicz-Sobolev空间中考察一类自由边界问题极小化子正则性理论.通过建立极小化问题与方程强解问题的等价性,建立二相障碍问题极小化子的存在唯一性及C^(1,α)-正则性.最后建立极小化子在自由边界附近的非退化性.

一类分数阶偏微分方程的正则性分析

讨论了一类分数阶对流-扩散方程,首先用分离变量法求出方程的解,然后用Adomian拆分法通过数值算例与分离变量法的结果进行比较,验证了方法的有效性.接着讨论了方程的强解、正则性、连续性等性质.

一类非线性拟抛物方程的初边值问题

考虑一类非线性拟抛物方程ｕｔ－ｕｘｘｔ＋ｆ（ｕ）－ｇ（ｕｘ）ｘ＝ｈ（ｘ，ｔ）的初边值问题．证明了整体强解的存在唯一性，并讨论了对应非负初值解的非负性、正则性、渐近性及爆破问题．

关于3维Navier－Stokes方程解的整体正则性研究

本文对３维Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程解的整体正则性进行了研究，得到了解在一些更弱的范数意义下的“小初值”正则性结果。

electro-hydrodynamics方程的弱L^p Prodi-Serrin型正则准则(英)

主要研究了electro-hydrodynamics方程弱解的正则准则.证明了在条件u∈L^s(0,T;L^(r,∞)(Ω))或‖u‖L^(s,∞)(0,T;L^(r,∞)(Ω))≤C(其中(3/r)+(2/s)=1且r∈(3,∞],C=C(r,Ω)>0)下,弱解在区间[0,T]上也为强解.

一类拟线性椭圆方程的强解的存在性

文章研究了一类二阶拟线性椭圆方程的Dirichlet问题.首先利用弱连续算子的锐角原理得到了此类方程的弱解存在性,然后运用极值原理,Hlder模估计及H1+α正则性估计对解的正则性进行了提升,最后通过Lp理论得到了强解的存在性和唯一性.

二阶线性椭圆型方程强解的Holder连续性

本文应用弱Harnack不等式及局部极大值原理,证明方程(1)的强解在区域Ω中的局部Holder连续性.

Parametric Anisotropic(p,q)-Neumann Problems

We consider a Neumann problem driven by a(p(z), q(z))-Laplacian(anisotropic problem) plus a parametric potential term with λ > 0 being the parameter. The reaction is superlinear but need not satisfy the Ambrosetti-Rabinowitz condition. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter λ moves on R_(+)=(0,+∞).

管道中三维不可压缩Navier-Stokes方程组解的正则性准则

本文研究了三维无限长管道中的不可压缩的Navier-Stokes方程组,根据u=(u_1,u_2,u_3)的导数建立了解的整体正则性判据.

The Existence and Uniqueness of a New Boundary Value Problem(Type of Problem“E”)for a Class of Semi Linear(Power Type Nonlinearities)Mixed Hyperbolic-Elliptic System Equations of Keldysh Type with Changing Time Direction

In present work studied the new boundary value problem for semi linear(Power-type nonlinearities)system equations of mixed hyperbolic-elliptic Keldysh type in the multivariate dimension with the changing time direction.Considered problem and equation belongs to the modern level partial differential equations.Applying methods of functional analysis,topological methods,“ε”-regularizing.and continuation by the parameter at the same time with aid of a prior estimates,under assumptions conditions on coefficients of equations of system,the existence and uniqueness of generalized and regular solutions of a boundary problem are established in a weighted Sobolev's space.In this work one of main idea consists of show that the new boundary value problem which investigated in case of linear system equations can be well-posed when added nonlinear terms to this linear system equations,moreover in this case constructed new weithged spaces,the identity between of strong and weak solutions is established.

不确定多目标优化的鲁棒解及强弱Kuhn-Tucker条件

【目的】对不确定多目标优化问题的鲁棒解及相关性质进行讨论。【方法】基于Burachik定义的两个正则条件,提出两个新的正则条件,即鲁棒性正则条件,进而利用新正则条件对不确定多目标优化问题的鲁棒解进行研究。【结果】建立了该问题鲁棒有效解的弱Kuhn-Tucker必要条件和真鲁棒有效解的强Kuhn-Tucker必要条件。【结论】所得的主要结果是对最近一些研究工作的改进和推广。