三维带有衰减项的不可压缩磁流体力学方程组弱解与强解的研究

论文研究了带有衰减项的磁流体力学方程组的柯西问题.当β≥1及初值u_0,b_0∈L~2(R~3)时,采用Galerkin方法证明了方程组存在全局弱解.并且当初值u_0∈H_0~1∩L~(β+1)(R~3),b_0∈H_0~1(R~3)时,可以得到方程组存在唯一局部强解.     

双线性Fourier乘子交换子的有界性与紧性

假定T_σ是关于乘子σ的双线性Fourier乘子算子,其中σ满足如下Sobolev正则条件:对某个s∈(n,2n],有sup_(κ∈Z)‖σ_k‖W~s(R~(2m))∞.对于p_1,p_2,p∈(1,∞)且满足1/p=1/p_1+1/p_2和ω=(ω_1,ω_2)∈A_(p/t)(R~(2n)),建立了T_σ及其与函数b=(b_1,b_2)∈(BMO(R~n))~2生成的交换子T_(σ,b)由L~(p_1,λ)(ω_1)×L~(p_2,λ)(ω_2)到L~(p,λ)(v_w)的有界性;同时,在b_1,b_2∈CMO(R~n)(C_c~∞(R~n)在BMO拓扑下的闭包)的条件下,证明交换子T_(σ,b)是L~(p_1,λ)(ω_1)×L~(p_2,λ)(ω_2)到L~(p,λ)(v_w)的紧算子.为了得到主要结果,我们先后建立了几个双(次)线性极大函数在加多权Morrey空间上的有界性以及该空间中准紧集的判定.     

有界区域上非线性 Schrodinger 方程解的全局性质

本文解决了 H.Bréis 在文[1]中提出的一个问题,对于非线性 Schr(?)dinger 方程(?) (1)H.Brézis 在文[1]中证明了,如果 Ω=R~2,p=3,K≤0或k>0,同时K integral from R~2|u_0|~2dx<4,(1)式有解 u(x,t),u∈C~1([0,∞),L~2(R~2))∩C([0,∞),H~2(R~2)).并且指出,在 Ω=R~2的情形,如果 p≥3,并且初值 u_0满足     

有界区域上非线性Schrdinger方程解的全局性质

本文解决了 H.Bréis 在文[1]中提出的一个问题,对于非线性 Schr(?)dinger 方程(?) (1)H.Brézis 在文[1]中证明了,如果 Ω=R~2,p=3,K≤0或k>0,同时K integral from R~2|u_0|~2dx<4,(1)式有解 u(x,t),u∈C~1([0,∞),L~2(R~2))∩C([0,∞),H~2(R~2)).并且指出,在 Ω=R~2的情形,如果 p≥3,并且初值 u_0满足     

ALMOST CONSERVATION LAWS AND GLOBAL ROUGH SOLUTIONS OF THE DEFOCUSING NONLINEAR WAVE EQUATION ON R~2

In this article, we investigate the initial value problem(IVP) associated with the defocusing nonlinear wave equation on R~2 as follows:{?_(ttu)-△u =-u~3,u(0, x) = u_0(x), ?_(tu)(0, x) = u_1(x),where the initial data(u_0, u_1) ∈ H~s(R~2) × H~(s-1)(R~2). It is shown that the IVP is global well-posedness in H~s(R~2) × H~(s-1)(R~2) for any 1 s 2/5. The proof relies upon the almost conserved quantity in using multilinear correction term. The main difficulty is to control the growth of the variation of the almost conserved quantity. Finally, we utilize linear-nonlinear decomposition benefited from the ideas of Roy [1].     

RESEARCH ANNOUNCEMENTS ON“MINIMAL TIME CONTROL OF EXACT SYNCHRONIZATION FOR PARABOLIC SYSTEMS”

正1 Introduction and Main Results LetΩ■R~d (with d≥1) be a bounded domain with a C~2 boundary Ω.Letω■Ωbe an open and nonempty subset with its characteristic function χ_ω.Let A■(a_(ij))_(1≤i,j≤n)∈R~(n×n)and B■(b_(ij))_(1≤i≤n,1≤j≤m)∈R~(n×m) be two constant matrices,where n≥2 and m≥1.Let y_0∈L~2(Ω)~n.Consider the controlled linear parabolic system     

对数平均和双参数广义Muirhead平均之间的比较

本文讨论了两个不同正实数x和y的对数平均L(x,y)=(x-y)/(logx-logy)与双参数广义Muirhead平均M(a,b;x,y)=[(x~ay~b+x~by~a)/2]~(1/(a+b))之间的比较,得到了如下三个结论:(11)若(a,b)∈D_1∪E_1∪L_0,则M(a,b;x,y)L(x,y);(2)若(a,b)∈D_2∪E_2,则M(a,b;x,y)L(x,y);(3)若(a,b)∈D_3∪E_3,则存在x_1,y_1,x_2,y_2,使得M(a,b;x_1;y_1)L(x_1,y_1)和M(a,b;x_2,y_2)L(x_2,y_2).其中D_1={(a,b)∈R~2:a+b≠0,ba,ω_1(a,b)≤0,ω_2(a,b)≤0},E_1={(a,b)∈R~2:a+b≠0,ba,ω_1(a,b)≤0,ω_2(a,b)≤0},D_2={(a,b)∈R~2:ab≤0,ba,ω_1(a,b)≥0},E_2={(a,b)∈R~2:ab≤0,ba,ω_1(a,b)≥0},D_3={(a,b)∈R~2:ba0,ω_1(a,b)0)∪{(a,b)∈R~2:ba0,ω_1(a,b)=0,ω_2(a,b)0}∪{(a,b)∈R~2:ba,ab≤0,ω_1(a,b)0,ω_2(a,b)0},E_3={(a,b)∈R~2:ab0,ω_1(a,b)0}∪{(a,b)∈R~2:ab0,ω_1(a,b)=0,ω_2(a,b)0}∪{(a,b)∈R~2:ab,ab≤0,ω_1(a,b)0,ω_2(a,b)0},L_0={(a,b)∈R~2:a=b≠0},ω_1(a,b)=(a+b)[3(a-b)~2-(a+b)],ω_2(a,b)=(a+b)[2(a-b)~2+1]-3(a~2+b~2).     

不定常耦合型偏微分方程组非协调元方法的稳定性和收敛性

(1)a_(ij)~(k)(x)充分光滑,A~(k)(x)对x∈Ω为一致正定、有界的对称矩阵。 (2)对(x,p)∈Ω×R~2,D_1(x,p),一致有界且关于p满足Lipschitz条件。对(x,t,p)∈Ω×[0,T]×R~2,F(x,t,p)对P满足Lipschitz条件,F(x,t,0)∈L~∞([0,T];L~2(Ω)×L~2(Ω))。     

索伯列夫空间的有限元迫近

记I_1=(-∞,ξ_1),I_2=(ξ_1,ξ_2),…,I_n=(ξ_(n-1),ξ_n),I_(n 1)=(ξ_n, ∞)。定义H~(m 1)(R,ξ_1,…,ξ_n)={u|u∈H~m(R),在I_i上u∈H~(m 1)(I_i),i=1,…,n 1}。 设μ(x)∈H~m(R),λ(x)∈L~∞(R)。并且满足:1.他们的支集都是R中的有界集合;2·∫_Rμ(x)dx=∫_Kλ(x)dx=1;3.μ(x)满足m-1收敛准则条件,即存在常数b_0=1,b_1,…,     

GLOBAL WELL-POSEDNESS IN ENERGY SPACE OF SMALL AMPLITUDE SOLUTIONS FOR KLEIN-GORDON-ZAKHAROV EQUATION IN THREE SPACE DIMENSION

The Cauchy problem of the Klein-Gordon-Zakharov equation in three dimensional space {utt-?u + u =-nu,(x, t) ∈ R~3× R_+,ntt-?n= ?|u|~2,(x, t) ∈ R~3× R_+,u(x, 0) = u_0(x), ?_tu(x, 0) = u_1(x),n(x, 0) = n_0(x), ?_tn(x,0) =n_1(x),(0.1) is considered. It is shown that it is globally well-posed in energy space H~1× L~2× L~2× H~(-1) if small initial data(u_0(x), u_1(x), n_0(x), n_1(x)) ∈(H~1× L~2× L~2× H~(-1)). It answers an open problem: Is it globally well-posed in energy space H~1× L~2× L~2× H~(-1) for 3D Klein-GordonZakharov equation with small initial data [1, 2]? The method in this article combines the linear property of the equation( dispersive property) with nonlinear property of the equation(energy inequalities). We mainly extend the spaces F~s and N~s in one dimension [3] to higher dimension.     

正阻尼摆方程的全局吸引性

本文考虑摆方程d~2x/dt~2+f(t,x)dx/dt+g(t,x)=0,这里f(t,x),g(t,x)∈C~1(R~1/ωZ~1×R~1/ω_1Z~1).我们给出一个充分条件保证上方程在其轨道空间(t,x,x)∈R~1/ωZ~1×R~1/ω_1Z~1×R~1中有唯一的不变环面,且方程的所有解都指数趋于此不变环面.     

正阻尼摆方程的全局吸引性

本文考虑摆方程d~2x/dt~2+f(t,x)dx/dt+g(t,x)=0,这里f(t,x),g(t,x)∈C~1(R~1/ωZ~1×R~1/ω_1Z~1).我们给出一个充分条件保证上方程在其轨道空间(t,x,x)∈R~1/ωZ~1×R~1/ω_1Z~1×R~1中有唯一的不变环面,且方程的所有解都指数趋于此不变环面.     

带非光滑核的多线性Marcinkiewicz积分的加权界

我们引入了带非光滑核的多线性Marcinkiewicz积分算子.设p_1,…,p_m∈(1,∞)和p∈(0,+∞)满足1/p_1+…+1/p_m=1/p,记P=(p_1,…,p_m),又设向量权ω=(ω_1,…,ω_m)∈A_p和v_ω=Π_(k=1)~mω_k~(p/pk),得到了Marcinkiewicz积分算子从L~(p_1)(ω_1)×…×L~(p_m)(ω_m)到L~p(v_ω)的常数界.     

On the bilinear square Fourier multiplier operators and related multilinear square functions

Let n 1 and Tm be the bilinear square Fourier multiplier operator associated with a symbol m,which is defined by Tm(f1, f2)(x) =(∫_0~∞︱∫_((Rn)2)e~(2πix·(ξ1+ξ2))m(tξ1, tξ2)?f1(ξ1)?f2(ξ2)dξ1dξ2︱~2(dt)/t) ~(1/2).Let s be an integer with s ∈ [n + 1, 2n] and p0 be a number satisfying 2n/s p0 2. Suppose that νω=∏_i~2=1ω_i~(p/pi) and each ω_i is a nonnegative function on Rn. In this paper, we show that under some condition on m, Tm is bounded from L~(p1)(ω_1) × L~(p2)(ω_2) to L~p(ν_ω) if p0 p1, p2 ∞ with 1/p = 1/p1 + 1/p2. Moreover,if p0 2n/s and p1 = p0 or p2 = p0, then Tm is bounded from L~(p1)(ω_1) × L~(p2)(ω_2) to L~(p,∞)(ν_ω). The weighted end-point L log L type estimate and strong estimate for the commutators of Tm are also given. These were done by considering the boundedness of some related multilinear square functions associated with mild regularity kernels and essentially improving some basic lemmas which have been used before.     

振动方程混合有限元方法的后处理

1引 言 考虑下面的振动方程混合问题 u_u+△~2u=f, (x,t)∈Ω×(0,T], u_1(x,0)=w_0,u(x,0)=u_0,x∈Ω, (1.1) u=u/γ=0, (x,t)∈Ω×(0,T],其中ΩR~2为有界规则区域,Ω为其逐段光滑的边界,u/γ表示u沿Ω的外法向导数,T>0为常数,f∈L~2(Ω)为已知函数。 引入涡度函数v=△u,则(1.1)改写为     

三维不可压Navier-Stokes方程关于速度场单分量在各向异性临界空间中的正则性准则 献给杨乐教授80华诞

给定初始涡度场ω_0属于L~(3/2)∩L~2,本文证明若三维不可压Navier-Stokes方程的Fujita-Kato解在有限时刻T~?处发生爆破,则对任意p∈[4,∞], q_1∈[1, 2],μ 0, q_2∈[2,(1/p+μ)~(-1)],κ∈[1,∞],以及任意单位向量e,(v(t)|e)_(R~3)的■范数等于无穷.     

关于广义Schrodinger型方程组第三边值问题的差分方法

本文讨论下述定解问题的差分解法 u_t(x,t)=Au_(xx)(x,t) f(u),(x,t)∈Q_T=(0,L)×(0,T) u_x(0,t)—σ_1u(0,t)=0,σ_1>0,t∈[0,T]; u_x(L,t) σ_2u(L,t)=0,σ_2>0,t∈[0,T]; u(x,0)=■(x),x∈[0,L].其中u(x,t)=(u_1(x,t),…,u_m(x,t)),f(u)=f(f_1(u),…,f_m(u)),■(x)=(■_1(x),…■_m(x))满足适定性条件,且假定     

高维广义BBM方程组的初边值问题

本文应用先验估计和Galerkin方法证明了高维广义BBM方程组的初边值问题在L~∞(0,T;H~3(Ω)∩H_0~1(Ω)),(s≥2)中整体解的存在性和正则性,并得到了整体解在||·||_(H~3×L~∞)范数下的稳定性和光滑解的唯一性。     

三维定常流Stokes问题的边界积分方程法

1.解的积分表示及变分公式 考虑如下Stokes方程的Dirichlet问题:设Ω是具有光滑边界Γ的单连通区域,Ω′=R~3-Ω。所求未知量是充满于Ω或Ω′的不可压缩粘性流体的流速u=(u_1,u_2,u_3)和压力p。这里v是运动粘性系数。 已经证明[Nedelec-Communication personnelle]:若u_0∈(H~(1/2)(Γ))~3,且满足     

带有幂型的广义Zakharov方程组解的爆破率的下界估计

主要研究了关于R~2中一类带有幂型非线性的广义Zakharov方程组的Cauchy问题的有限时间爆破解的爆破率的下界估计.在α≤0和p≥3条件下,对于Cauchy问题任意给定的属于能量空间H~1(R~2)×L~2(R~2)×L~2(R~2)的有限时间的爆破解,得到了对于t靠近有限爆破时间T时的爆破率的最优下界估计.此外,给出了Cauchy问题维里等式的一个应用.