带势非线性Schrdinger方程爆破解的集中性质

本文研究带排斥调和势非线性schrdinger方程的爆破解.我们得到如下精细的集中性质:如果初值条件满足‖u_0‖_(L~2)=‖Q‖_(L~2),那么方程对应的爆破解满足:当t→T时,|u(t,x)|~2→‖Q‖_(L~2)~2δ_x=x_1,其中T为爆破时刻.此外,我们还讨论了方程爆破解在其能量空间∑:={ν∈H~1(R~N)||x|v∈L~2(R~N))中的极限行为.     

二维空间中一类具临界幂的耦合非线性波动系统的爆破解

在二维空间中建立了一类具临界幂的非线性耦合波动系统爆破解的存在性.并进一步研究爆破解的定性行为,得到爆破解的L~2集中性质.     

弱于H~1空间的L~2-临界非线性Schrdinger方程的柯西问题

讨论H~s(R~n)(n≥1,1-ε0是一个可以表出的很小的数.主要结论给出了在有限时间破裂解的L~2集中现象.同时,作为推论,得到了小初值解的整体存在性.     

一类N维非线性波动方程的Cauchy问题

证明下列非线性波动方程的Cauchy问题v_(tt)-α△v_(tt)-Δv=g(v)-αΔg(v),x∈R~N,t>0,(1)v(x,0)=v_0(x),v_t(x,0)=v_1(x),x∈R~N(2)在空间C~2([0,∞);H~s(R~N))(s>N/2)中存在唯一整体广义解v和在空间C~2([0,∞);H~s(R~N))(s>2+N/2N)中存在唯一整体古典解v,即u∈C~2([0,∞);C_B~2(R~N)).还证明Cauchy问题(1),(2)在C~3([0,∞);W~(m,p)(R~N)∩L~∞(R~N))(m≥0,1≤p≤∞)中有唯一整体广义解v和在C~3([0,∞);W~(m,p)(R~N)∩L~∞(R~N))(m>2+N/P)中有唯一整体古典解v,即v∈C~3([0,∞);C~2(R~N)∩L~∞(R~N)).     

有限区间上的Korteweg-de Vries方程的快速镇定

Coron和L(2014)研究了区间(0,L)上的Korteweg-de Vries方程所描述的一类控制系统的快速镇定问题.对λ∈(0,∞),他们设计出了适当的状态反馈并证明了相应的闭环系统△_(λ,L)在L~2(0,L)中局部适定,闭环系统△_(λ,L)的状态轨线在L~2(0,L)中呈λ/2型指数衰减.本文进一步证明对_s∈[0,∞)\{j+1/2;j∈N_0}:(i)闭环系统△_(λ,L)在H~s(0,L)中整体适定;(ii)闭环系统△_(λ,L)的状态轨线在H~s(0,L)中仍然呈局部的λ/2型指数衰减.     

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.     

一类具有局部化源和吸收项的抛物系统解的整体存在与爆破

在齐次Dirichlet边界条件研究如下抛物系统其中x_0(t):R⁺→(0,a)是Holder连续函数;常数0≤α,β<1,p_1,p_2,q_1,q_2,k_1,k_2>0.利用正则化方法,在一定的假设条件下证明了经典解的存在性.接着利用比较原理证明了该系统正解的整体存在性和爆破性.最后给出了爆破解的精确爆破速率和爆破模式.     

带部分调和势的非齐次非线性Schr?dinger方程的爆破解

该文致力于研究带部分调和势的非齐次非线性Schr?dinger方程的Cauchy问题.该方程是玻色-爱因斯坦凝聚中的一个重要模型.结合非线性椭圆方程基态解的变分特征及质量和能量守恒,首先得到了该问题整体解的存在性,并利用尺度变换技巧证明了该方程在一些特殊初值情形下存在爆破解.其次讨论了爆破解的L²集中现象.最后利用与上述基态解相关的变分结论研究了L²最小质量爆破解的动力学性质,即具有最小质量的爆破解的极限profile、精细质量集中和爆破速率.该文将Zhang^[35]的全局存在性和爆破结果推广到带非齐次非线性项的情形,并将Pan和Zhang^[24]的部分结果改进到空间维数N≥2且非线性项为非齐次的情形.     

具有弱阻尼项的广义IMBq方程初值问题解的整体存在性

证明具有弱阻尼项的广义IMBq方程乱u_(tt)-u_(xx)-u_(xxtt)+v_0u_t=f(u)_(xx),x∈R,t0的初值问题在C~2([0,∞);H~s(R))(s2是一实数)中存在惟一整体广义解和在C~2([0,∞);H~s(R))(s2/7)中存在惟一整体古典解,并给出上述初值问题解爆破的充分条件.     

一类Capillarity系统非平凡解的存在性研究

本文研究了一类capillarity系统解的存在性问题.采用在乘积空间中定义非线性映射的方法,把capillarity系统转化为非线性算子方程.借助于Sobolev嵌入定理等技巧证明非线性映射具有紧性,进而利用非线性映射值域的性质得到非线性算子方程解的存在性的结论.并由此获得在一定条件下capillarity系统在L~(P1)(Ω)×L~(P2)(Ω)×…×L~(PM)(Ω)空间中存在非平凡解的结论,其中Ω为R~N(N≥1)中有界锥形区域且2N/N+1p_i+∞,i=1,2,…,M.本文所研究的问题和所采用的方法推广和补充了以往的相关研究工作.     

Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation

We consider the critical nonlinear Schrödinger equation $iu_{t} = -\Delta u-|u|^{4/N}$ with initial condition u(0, x) = u₀.For u₀$\in$H¹, local existence in time of solutions on an interval [0, T) is known, and there exist finite time blow-up solutions, that is u₀ such that $\textrm{lim} _{t\uparrow T <+\infty}|\nabla u(t)|_{L^{2}}=+\infty$. This is the smallest power in the nonlinearity for which blow-up occurs, and is critical in this sense.The question we address is to control the blow-up rate from above for small (in a certain sense) blow-up solutions with negative energy. In a previous paper [MeR], we established some blow-up properties of (NLS) in the energy space which implied a control $|\nabla u(t)|_{L^{2}} \leq C \frac{|\ln(T-t)|^{N/4}}{\sqrt{T-t}}$ and removed the rate of the known explicit blow-up solutions which is $\frac{C}{T-t}$.In this paper, we prove the sharp upper bound expected from numerics as$|\nabla u(t)|_{L^{2}} \leq C \left(\frac{\ln|\ln(T-t)|}{T-t} \right)^{1/2}$by exhibiting the exact geometrical structure of dispersion for the problem.     

On a sharp lower bound on the blow-up rate for the critical nonlinear Schrödinger equation

We consider the critical nonlinear Schrödinger equation with initial condition in the energy space and study the dynamics of finite time blow-up solutions. In an earlier sequence of papers, the authors established for a certain class of initial data on the basis of dispersive properties in a sharp and stable upper bound on the blow-up rate: .

In an earlier paper, the authors then addressed the question of a lower bound on the blow-up rate and proved for this class of initial data the nonexistence of self-similar solutions, that is,

In this paper, we prove the sharp lower bound

by exhibiting the dispersive structure in the scaling invariant space for this log-log regime. In addition, we will extend to the pure energy space a dynamical characterization of the solitons among the zero energy solutions.

     

An Adaptive Mesh-Refining Algorithm Allowing for an <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript> Stable <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> Projection onto Courant Finite Element Spaces

Suppose $\cal{S}^1({\cal T})\subset H^1(\Omega)$ is the $P_1$-finite element space of $\cal{T}$-piecewise affine functions based on a regular triangulation $\cal{T}$ of a two-dimensional surface $\Omega$ into triangles. The $L^2$ projection $\Pi$ onto $\cal{S}^1(\cal{T})$ is $H^1$ stable if $\norm{\Pi v}{H^1(\Omega)}\le C\norm{v}{H^1(\Omega)}$ for all $v$ in the Sobolev space $H^1(\Omega)$ and if the bound $C$ does not depend on the mesh-size in $\cal{T}$ or on the dimension of $\cal{S}^1(\cal{T})$. \hskip 1em A red–green–blue refining adaptive algorithm is designed which refines a coarse mesh $\cal{T}_0$ successively such that each triangle is divided into one, two, three, or four subtriangles. This is the newest vertex bisection supplemented with possible red refinements based on a careful initialization. The resulting finite element space allows for an $H^1$ stable $L^2$ projection. The stability bound $C$ depends only on the coarse mesh $\cal{T}_0$ through the number of unknowns, the shapes of the triangles in $\cal{T}_0$, and possible Dirichlet boundary conditions. Our arguments also provide a discrete version $\norm{h_\cal{T}^{-1}\,\Pi v}{L^2(\Omega)}\le C\norm{h_\cal{T}^{-1}\,v}{L^2(\Omega)}$ in $L^2$ norms weighted with the mesh-size $h_\T$.     

A NEW REGULARITY CLASS FOR THE NAVIER-STOKES EQUATIONS IN IR~n

ＡＮＥＷＲＥＧＵＬＡＲＩＴＹＣＬＡＳＳＦＯＲＴＨＥＮＡＶＩＥＲ－ＳＴＯＫＥＳＥＱＵＡＴＩＯＮＳＩＮＩＲ~ｎ￥Ｈ．ＢＥＩＲａＯＤＡＶＥＩＧＡ（ＤｅｐｏｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＰｉｓａＵｎｉｖｅｒｓｉｔｙ，Ｐｉｓａ，Ｉｔａｌｙ）Ａｂｓｔｒａ?..     

Complicated Asymptotic Behavior of Solutions for the Cauchy Problem of Doubly Nonlinear Diffusion Equation

In this paper, we analyze the large time behavior of nonnegative solutions to the doubly nonlinear diffusion equation $$u_t−{\rm div}(|∇u^m|^{p−2}∇u^m)=0$$ in $\mathbb{R}^N$ with $p>1,$ $m>0$ and $m(p−1)−1>0.$ By using the finite propagation property and the $L^1-L^∞$ smoothing effect, we find that the complicated asymptotic behavior of the rescaled solutions $t^{\mu/2}u(t^{β_·},t)$ for $0<\mu<2N/(N[m(p−1)−1]+p)$ and $β>(2−\mu[m(p−1)−1])/(2p)$ can take place.     

沿超曲面的强奇异积分算子在调幅空间上的有界性

设■该文主要讨论了上述奇异积分算子在广义的调幅空间上的有界性,其中粗糙核Ω∈L~1(S~(n-2))h(y)为有界的径向函数,而γ(y)是满足一定条件的超曲面.     

Entropy solutions to nonlinear elliptic anisotropic problem with variable exponent

In this work, we give an existence result of entropy solutions for nonlinear anisotropic elliptic equation of the type $$- \mbox{div} \big( a(x,u,\nabla u)\big)+ g(x,u,\nabla u) + |u|^{p_{0}(x)-2}u = f-\mbox{div} \phi(u),\quad \mbox{ in } \Omega,$$ where $-\mbox{div}\big(a(x,u,\nabla u)\big)$ is a Leray-Lions operator, $\phi \in C^{0}(I\!\!R,I\!\!R^{N})$. The function $g(x,u,\nabla u)$ is a nonlinear lower order term with natural growth with respect to $|\nabla u|$, satisfying the sign condition and the datum $f$ belongs to $L^1(\Omega)$.     

Sampling expansions for functions having values in a Banach space

A sampling expansion for vector-valued functions having values in a Banach space, together with an inversion formula, is derived. The proof uses the concept of framing models of Banach spaces that generalizes the notion of frames in Hilbert spaces. Two examples illustrating the results are given, one involving functions having values in , and the second involving functions having values in for

     

生成元一致连续的多维倒向随机微分方程的$L^p$解

在生成元~$g$~的第~$i$~个分量~$g_i(t,y,z)$~仅仅依赖于矩阵~$z$~的第~$i$ 行的条件下, Hamad\`{e}ne于2003年证明了生成元一致连续的倒向随机微分 方程的~$L^2$~解的存在性, 其~$L^2$~解的唯一性由范胜君等于2010年得到. 本文进一步地证明了该类倒向随机微分 方程的~$L^p\ (p>1)$~解的存在唯一性.