光滑度量测度空间上的加权扩散方程的梯度估计

本文主要考虑了一类加权非线性扩散方程正解的梯度估计.在m-维Bakry-(E)mery Ricci曲率下有界的假设下,得到加权多孔介质方程(γ＞1)正解的Li-Yau型梯度估计,此外对于加权快速扩散方程(0＜γ＜1),证明了Hamilton型椭圆梯度估计,结论分别推广了Lu,Ni,Vázquez and Villani在文[1]和Zhu在文[2]中的结果.     

GRADIENT ESTIMATES FOR POSITIVE SMOOTH f-HARMONIC FUNCTIONS

For Riemannian manifolds with a measure,we study the gradient estimates for positive smooth f-harmonic functions when the ∞-Bakry-Emery Ricci tensor and Ricci tensor are bounded from below,generalizing the classical ones of Yau(i.e.,when f is constant).     

ELLIPTIC GRADIENT ESTIMATES FOR DIFFUSION OPERATORS ON COMPLETE RIEMANNIAN MANIFOLDS

In this note,we obtain the elliptic estimate for diffusion operator L=△+φ on complete,noncompact Riemannian manifolds,under the curvature condition C D(K,m),which generalizes B.L.Kotschwar's work [5].As an application,we get estimate on the heat kernel.The Bernstein-type gradient estimate for Schro¨dinger-type gradient is also derived.     

Calderon-Zygmund型算子在加权Herz型Hardy空间上的弱型估计

本文引入了加权弱Herz型Hardy空间，并证明了当a=n(1-1/q) δ时，胡国恩和陆善镇等在文[1]中所考虑的两类Calderon-Zygmund型算子分别连续地映加权Herz型Hardy空间到加权弱Herz空间和加权弱Herz型Hardy空间。     

INTERIOR ESTIMATES IN MORREY SPACES FOR SOLUTIONS OF ELLIPTIC EQUATIONS AND WEIGHTED BOUNDEDNESS FOR COMMUTATORS OF SINGULAR INTEGRAL OPERATORS

It is proved that, for the nondivergence elliptic equations ∑in, j=1 aijuxixj = f,if f belongs to the generalized Morrey spaces Lp,ψ(ω), then uxixj ∈ Lp,ψ(ω), where u is the W2,p-solution of the equations. In order to obtain this, the author first establish the weighted boundedness for the commutators of some singular integral operators on Lp,ψ (ω).     

Gradient estimates and entropy monotonicity formula for doubly nonlinear diffusion equations on Riemannian manifolds

In the first part of this paper, we prove the sharp global Li‐Yau type gradient estimates for positive solutions to doubly nonlinear diffusion equation(DNDE) on complete Riemannian manifolds with nonnegative Ricci curvature. As an application, one can obtain a parabolic Harnack inequality. In the second part, we obtain a Perelman‐type entropy monotonicity formula for DNDE on compact Riemannian manifolds with nonnegative Ricci curvature. These results generalize some works of Ni (JGA 2004), Lu–Ni–Vázquez–Villani (JMPA 2009) and Kotschwar–Ni (Annales Scientifiques de l'École Normale Supérieure 2009). Copyright © 2013 John Wiley & Sons, Ltd.     

加权Bergman空间上的Carleson测度与复合算子

本文研究了复平面上单位圆盘D上具有指数型权的加权Bergman空间上的Carleson测度和复合算子。利用Carleson测度的概念分别给出了有界或紧的复合算子的充分必要条件。     

GRADIENT ESTIMATES AND LIOUVILLE THEOREMS FOR LINEAR AND NONLINEAR PARABOLIC EQUATIONS ON RIEMANNIAN MANIFOLDS

In this article, we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations(?-?/(?t))u(x, t) + h(x,t)u(x,t) = 0 and nonlinear parabolic equations(?-?-/(?t))u(x,t) + h(x, t)u~p(x,t) = 0(p 1) on Riemannian manifolds.As applications, we obtain some theorems of Liouville type for positive ancient solutions of such equations. Our results generalize that of Souplet-Zhang([1], Bull. London Math. Soc.38(2006), 1045-1053) and the author([2], Nonlinear Anal. 74(2011), 5141-5146).     

L_p-L_q DECAY ESTIMATES FOR HYPERBOLIC EQUATIONS WITH OSCILLATIONS IN COEFFICIENTS

51.IntroductionToproveglobalealstenceresultsforthesolutionsoftheCauchyproblemfornonlinearwaveequationsso-calledL.--L,decayestimatesforthesollltionsofthelinearwaveequationplananessentialr.l.[3'4'7].ThatisthefollowingestimateduetoSt.i.hatz[12]:thereedestconstantsCandLdependingonpandnsuchthatwhere1相似文献   

A STUDY ON GRADIENT BLOW UP FOR VISCOSITY SOLUTIONS OF FULLY NONLINEAR, UNIFORMLY ELLIPTIC EQUATIONS

We investigate sharp conditions for boundary and interior gradient estimates of continuous viscosity solutions to fully nonlinear,uniformly elliptic equations under Dirichlet boundary conditions. When ...     

A NATURAL GRADIENT ALGORITHM FOR THE SOLUTION OF LYAPUNOV EQUATIONS BASED ON THE GEODESIC DISTANCE

A new framework based on the curved Riemannian manifold is proposed to calculate the numerical solution of the Lyapunov matrix equation by using a natural gradient descent algorithm and taking the geodesic distance as the objective function. Moreover, a gradient descent algorithm based on the classical Euclidean distance is provided to compare with this natural gradient descent algorithm. Furthermore, the behaviors of two proposed algorithms and the conventional modified conjugate gradient algorithm are compared and demonstrated by two simulation examples. By comparison, it is shown that the convergence speed of the natural gradient descent algorithm is faster than both of the gradient descent algorithm and the conventional modified conjugate gradient algorithm in solving the Lyapunov equation.     

PARALLEL SCHWARZ ALGORITHMS FOR PARABOLIC EQUATIONS AND ERROR ESTIMATES

In this paper we introduce two kinds of parallel Schwarz domain decomposition me thods for general, selfadjoint, second order parabolic equations and study the dependence of their convergence rates on parameters of time-step and space-mesh. We prove that the, approximate solution has convergence independent of iteration times at each time-level. And the L~2 error estimates are given.     

DECAY ESTIMATES OF PLANAR STATIONARY WAVES FOR DAMED WAVE EQUATIONS WITH NONLINEAR CONVECTION IN MULTI-DIMENSIONAL HALF SPACE

This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the multi-dimensional half space R n + : u tt u + u t + divf (u) = 0, t > 0, x = (x 1 , x ′ ) ∈ R n + := R + × R n 1 , u(0, x) = u 0 (x) → u + , as x 1 → + ∞ , u t (0, x) = u 1 (x), u(t, 0, x ′ ) = u b , x ′ = (x 2 , x 3 , ··· , x n ) ∈ R n 1 . (I) For the non-degenerate case f ′ 1 (u + ) < 0, it was shown in [10] that the above initialboundary value problem (I) admits a unique global solution u(t, x) which converges to the corresponding planar stationary wave φ(x 1 ) uniformly in x 1 ∈ R + as time tends to infinity provided that the initial perturbation and/or the strength of the stationary wave are sufficiently small. And in [10] Ueda, Nakamura, and Kawashima proved the algebraic decay estimates of the tangential derivatives of the solution u(t, x) for t → + ∞ by using the space-time weighted energy method initiated by Kawashima and Matsumura [5] and improved by Nishihkawa [7]. Moreover, by using the same weighted energy method, an additional algebraic convergence rate in the normal direction was obtained by assuming that the initial perturbation decays algebraically. We note, however, that the analysis in [10] relies heavily on the assumption that f ′ (u) < 0. The main purpose of this paper isdevoted to discussing the case of f ′ 1 (u b ) ≥ 0 and we show that similar results still hold for such a case. Our analysis is based on some delicate energy estimates.     

带振荡核奇异积分算子交换子在加权Morrey空间中的有界性质

本文研究Morrey空间中的交换子有界性的问题.利用John-Nirenberg不等式等工具建立带振荡核的奇异积分算子与BMO函数生成交换子在加权Morrey空间中的加权估计.     

REMARK ON THE BLOW-UP CRITERION OF STRONG SOLUTIONS TO THE NAVIER-STOKES EQUATIONS IN MULTIPLIER SPACES

In this note, we prove that Xr （0 〈 r 〈 1） norm of the vorticity controls the blow-up phenomena of strong solutions to the Navier-Stokes equations in R3.     

SUPERCONVERGENCE AND A POSTERIORI ERROR ESTIMATES FOR BOUNDARY CONTROL GOVERNED BY STOKES EQUATIONS

In this paper, the superconvergence results are derived for a class of boundary con-trol problems governed by Stokes equations. We derive superconvergence results for boththe control and the state approximation. Base on superconvergence results, we obtainasymptotically exact a posteriori error estimates.     

GLOBAL ATTRACTOR AND ITS DIMENSION ESTIMATES FOR THE GENERALIZED DISSIPATIVE KdV EQUATION ON R

1.IntroductionIn[41,GhidagliaprovedthatforthedissipativeKdVequation:withperiodicboundaryconditionu(x,t)=u(x L,t),thereexistsaweakglobalattractoroffinitedimension.Recently,[7]obtainedasimilarresultfortheperiodicboundaryproblemwhereor>0,g/0,7>0,fEC'(R),If15A]ul'--',6>0,Gi(0,0)=0,i=1,2andforevery(CI,'2)eR',However,feweffortsaredevotedtotheexistenceofglobalattractorinunboundeddomain.Themaindifficultinthiscaseliesinthatthelinearprincipleoperatorisnothavecompactresolventoperatorandthuswecannot…     

GLOBAL EXISTENCE AND CONVERGENCE RATES OF SMOOTH SOLUTIONS FOR THE 3-D COMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS WITHOUT HEAT CONDUCTIVITY

In this paper, we are concerned with the global existence and convergence rates of the smooth solutions for the compressible magnetohydrodynamic equations without heat conductivity, which is a hyperbolic-parabolic system. The global solutions are obtained by combining the local existence and a priori estimates if H3-norm of the initial perturbation around a constant states is small enough and its L1-norm is bounded. A priori decay-in-time estimates on the pressure, velocity and magnetic field are used to get the uniform bound of entropy. Moreover, the optimal convergence rates are also obtained.     

椭圆型方程的重叠型区域分裂混合元方法

本文研究椭圆型方程的重叠型区域分解混合元方法，对第一边值和第二边值问题，分别给出了离散形式的区域分解混合元格式；证明了区域分裂格式解的存在唯一性和算法的收敛性，并给出数值算例．