Regularity Criterion of Weak Solutions to the 3D Incompressible Hall-magnetohydrodynamic Equations

In this paper,we study the regularity criterion of weak solutions to the3 D incompressible Hall-magnetohydrodynamics,which is ifu and Bsatisfy the condition∫_0^T‖_（x3）u（t）‖^q_（LP）＋‖▽B‖^γ_（Lβ）dt〈∞ with 3/p＋2/q≤1,3/β＋2/γ≤1,p〉3,β〉3,then the weak solution（u,B） is a smooth one on（0,T].

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）.

The Smoothness of Weak Solutions to the System of Second Order Differential Equations with Non-negative Characteristics

In this paper,we will discuss smoothness of weak solutions for the system of second order differential equations eith non-negative characteristies.First of all,we establish boundary,and interior estimates and then we prove that solutions of regularization problem satisfy Lipschitz condition.

REGULARITY OF WEAK SOLUTIONS TO A CLASS OF NONLINEAR PROBLEM

We study the regularity of weak solutions to a class of second order parabolic system under only the assumption of continuous coefficients.We prove that the weak solution u to such system is locally Holder continuous with any exponent α∈(0,1)outside a singular set with zero parabolic measure.In particular,we prove that the regularity point in Q_(T) is an open set with full measure,and we obtain a general criterion for the weak solution to be regular in the neighborhood of a given point.Finally,we deduce the fractional time and fractional space differentiability of D_(u),and at this stage,we obtain the Hausdorff dimension of a singular set of u.

REGULARITY OF H^1 ∩L^(n((γ-1)/(2-γ))) WEAK SOLUTIONS FOR NONLINEAR ELLIPTIC SYSTEMS

Let us consider the following elliptic systems of second order-D_α(A_i~α(x, u, Du))=B_4(x, u, Du), i=1, …, N, x∈Q(?)R^n, n≥3 (1) and supposeⅰ) |A_i~α(x, u, Du)|≤L(1+|Du|);ⅱ) (1+|p|)^(-1)A_i~α(x, u, p)are H(?)lder-continuous functions with some exponent δ on (?)×R^N uniformly with respect to p, i.e.ⅲ) A_i~α(x, u, p) are differentiable function in p with bounded and continuous derivativesⅳ)ⅴ) for all u∈H_(loc)~1(Ω, R^N)∩L^(n(γ-1)/(2-γ))(Ω, R^N), B(x, u, Du)is ineasurable and |B(x, u, p)|≤a(|p|~γ+|u|~τ)+b(x), where 1+2/n<γ<2, τ≤max((n+2)/(n-2), (γ-1)/(2-γ)-ε), (?)ε>0, b(x)∈L2n/(n+2), n^2/(n+2)+e(Ω), (?)ε>0.Remarks. Only bounded open set Q will be considered in this paper; for all p≥1, λ≥0, which is clled a Morrey Space.Let assumptions ⅰ)-ⅳ) hold, Giaquinta and Modica have proved the regularity of both the H^1 weak solutions of (1) under controllable growth condition |B|≤α(|p|~γ+|u|^((n+2)/(n-2))+b, 0<γ≤1+2/n and the H^1∩L~∞ weak solutions of (1) under natural growth condition |B|≤α|p|~2+b with a smallness condition 2aM<λ(|u|≤M), which implys that the H^1∩L~∞ weak solutions have the same regularty in the case of 1+2/n<γ<2. In the case of γ=2, many counterexamples (see [2] showed that u must be in H^1L~∞, while in the case of 1+2/n<γ<2, we consider the H^1∩L^n(γ-1)/(2-γ) weak solutions of (1), weaken the instability conditions upon them (from L~∞ to L^n(γ-1)/(2-γ) and obtain the same regularity results. Finally we show that the exponent n(γ-1)/(2-γ) can not be docreased anymore for the sake of the regularity results.Delinition 1. We call u∈H^1∩L^n(γ-1)/(2-γ)(Q, R^N) be a weak solution of (1), providod that where We use the convention that repeated indices are summed. i, j go from 1 to N ann α, β from 1 to n.

GLOBAL WEAK SOLUTIONS TO THE α-MODEL REGULARIZATION FOR 3D COMPRESSIBLE EULER-POISSON EQUATIONS

Global in time weak solutions to the α-model regularization for the three dimensional Euler-Poisson equations are considered in this paper. We prove the existence of global weak solutions to α-model regularization for the three dimension compressible EulerPoisson equations by using the Fadeo-Galerkin method and the compactness arguments on the condition that the adiabatic constant satisfies γ >4/3.

PARTIAL REGULARITY FOR THE NAVIER-STOKES-FOURIER SYSTEM

This paper addresses a nonstationary flow of heat-conductive incompressible Newtonian fluid with temperature-dependent viscosity coupled with linear heat transfer with advection and a viscous heat source term, under Navier/Dirichlet boundary conditions. The partial regularity for the velocity of the fluid is proved for each proper weak solution, that is, for such weak solutions which satisfy some local energy estimates in a similar way to the suitable weak solutions of the Navier-Stokes system. Finally, we study the nature of the set of points in space and time upon which proper weak solutions could be singular.

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).

The Regularity of the Weak Solutions for the N-Dimensional Quasilinear Elliptic Equations with Discontinuous Coefficients

The n-dimensional quasilinear elliptic equations with discontinuous coefficients are studied. Using estimate and difference approach methods, we prove that the first derivatives of the weak solutions are continuous in the sense of Hlder up to the inner boundary on which the coefficients are discontinuous.

L^q Interior Estimates of the Gradient for Weak Solutions of Quasilinear Elliptic Equations

The authors present a new proof of L^q interior estimates of the gradient for weak solutions of a quasilinear elliptic equation of p-Laplacian type.

On Partial Regularity of Suitable Weak Solutions to the Stationary Fractional Navier-Stokes Equations in Dimension Four and Five

In this paper, we investigate the partial regularity of suitable weak solutions to the multi- dimensional stationary Navier-Stokes equations with fractional power of the Laplacian （-△）^α （n/6 ≤α〈1 and a ≠ 1/2）. It is shown that the n ＋ 2 - 6α （3 ≤ n ≤5） dimensional Hausdorff measure of the set of the possible singular points of suitable weak solutions to the system is zero, which extends a recent result of Tang and Yu [19] to four and five dimension. Moreover, the pressure in ε-regularity criteria is an improvement of corresponding results in [1, 13, 18, 20].

Partial Regularity of Suitable Weak Solutions to the System of the Incompressible Shear-thinning Flow

This paper is devoted to the partial regularity of suitable weak solutions to the system of the incompressible shear-thinning flow in a bounded domainΩ■R^(n),n≥2.It is proved that there exists a suitable weak solution of the shear-thinning fluid in the n-D smooth bounded domain(for n≥2).For 3 D model,it is proved that the singular points are concentrated on a closed set whose 1 dimensional Hausdorff measure is zero.

On the inviscid limit of the compressible Navier-Stokes equations near Onsager's regularity in bounded domains

The viscous dissipation limit of weak solutions is considered for the Navier-Stokes equations of compressible isentropic flows confined in a bounded domain.We establish a Kato-type criterion for the validity of the inviscid limit for the weak solutions of the Navier-Stokes equations in a function space with the regularity index close to Onsager’s critical threshold.In particular,we prove that under such a regularity assumption,if the viscous energy dissipation rate vanishes in a boundary layer of thickness in the order of the viscosity,then the weak solutions of the Navier-Stokes equations converge to a weak admissible solution of the Euler equations.Our approach is based on the commutator estimates and a subtle foliation technique near the boundary of the domain.

Existence and Regularity of a Weak Solution to a Class of Systems in a Multi-Connected Domain

We consider the existence and regularity of a weak solution to a class of systems containing a p-curl system in a multi-connected domain. This paper extends the result of the regularity theory for a class containing a p-curl system that is given in the author's previous paper. The optimal C^1+a-regularity of a weak solution is shown in a multi-connected domain.

PARTIAL REGULARITY FOR WEAK SOLUTIONS OF QUASILINEAR ELLIPTIC SYSTEMS

In this note, we study the partial regularity for the weak solutions of the elliptic systems:Dα(Aαβij（x,u）Dβuj)=fi（x,u,Du）, x∈Ω,i=1,2,…,N, （1）where Ω is a bounded domain in Rn, n≥3 and N≥1. Here, the repeated Latin letters andrepeated Greek letters are summed from 1 to N and 1 to n respectively. We assume thefollowing conditions:

Smoothness of the Gradient of Weak Solutions of Degenerate Linear Equations

Let Q（x） be a nonnegative definite, symmetric matrix such that √Q（X） is Lipschitz con- tinuous. Given a real-valued function b（x） and a weak solution u（x） of div（QVu） = b, we find sufficient conditions in order that √Qu has some first order smoothness. Specifically, if is a bounded open set in Rn, we study when the components of vVu belong to the first order Sobolev space W1＇2（Ω） defined by Sawyer and Wheeden. Alternately we study when each of n first order Lipschitz vector field derivatives Xiu has some first order smoothness if u is a weak solution in Ω of ^-^-1 X^Xiu ＋ b = O. We do not assume that {Xi}is a HSrmander collection of vector fields in ~. The results signal ones for more general equations.

Global existence of weak solutions to the drift-flux system for general pressure laws

The initial value problem of the multi-dimensional drift-flux model for two-phase flow is investigated in this paper,and the global existence of weak solutions with finite energy is established for general pressure-density functions without the monotonicity assumption.

A Regularity Criterion in Terms of Direction of Vorticity to the 3D Micropolar-fluid Equations

In this paper, we study the regularity of weak solutions to the 3D Micropolarfluid equations. We show that the weak solutions actually is strong solution if the corresponding vorticity field j = × u satisfies certain condition in the high vorticity region.

WEAK TRAVELLING WAVE FRONT SOLUTIONS OF GENERALIZED DIFFUSION EQUATIONS WITH REACTION

The author demonstrate that the two-point boundary value problemhas a solution (A,P(8)), where III is the smallest parameter, under the minimal stringent resstrictions oil f(8), by applying the shooting and regularisation methods. In a classic paper)Kolmogorov et. al. studied in 1937 a problem which can be converted into a special case of theabove problem.The author also use the solutioll (A, p(8)) to construct a weak travelling wave front solutionu(x, t) = y((), (= x -- Ct, C = AN/(N + 1), of the generalized diffusion equation with reactionO { 1 O.IN￣1 OUI onde L k(u) i ox: &) ￣ & = g(u),where N > 0, k(8) > 0 a.e. on [0, 1], and f(s):= ac i: g(t)kl/N(t)dt is absolutely continuouson [0, 11, while y(() is increasing and absolutely continuous on (--co, +co) and(k(y(())ly,(OI'), = g(y(()) -- Cy'(f) a.e. on (--co, +co),y(￣oo) ￣ 0, y(+oo) ￣ 1.

一类非线性发展方程解的长时间行为

研究一类非线性发展方程整体弱解的长时间行为,首先利用算子分解方法证明了系统整体弱解的渐近正则性,由此得到了整体弱解对应的解半群在H1(R3)×H1(R3)的全局吸引子A的存在性,然后证得A在H2(R3)×H2(R3)有界,其中非线性项满足临界指数增长.