Energy Scattering Theory for Electromagnetic NLS in Dimension Two

We study the well-posedness and long-time behavior of solution to both defocusing and focusing nonlinear Schr?dinger equations with scaling critical magnetic potentials in dimension two.In the defocusing case, and under the assumption that the initial data is radial, we prove interaction Morawetz-type inequalities and show the scattering holds in the energy space. The magnetic potential considered here is the Aharonov–Bohm potential which decays likely the Coulomb potential |x|~(-1).     

四阶正则化YK方程弱解的存在唯一性

该文针对图像处理中YK方程理论上的不适定性, 提出了正则化的YK方程, 证明了正则化的YK方程在有界区域上弱解的存在唯一性.而且, 该文只要求初始数据属于L², 这在图像处理中显得更为自然.     

On a Critical Fourth Order PDE with Navier Boundary Condition

We consider a fourth order nonlinear PDE involving the critical Sobolev exponent on a bounded domain of Rⁿ, n ≥ 5 with Navier condition on the boundary. We study the lack of compactness of the problem and we provide an existence theorem through a new index formula.     

Scattering for focusing combined power-type NLS

We consider the scattering of Cauchy problem for the focusing combined power-type Schr¨odinger equation. In the spirit of concentration-compactness method, we will show that, H1 solution will scatter under some condition on its energy and mass. We adapt some variance argument, following the idea of Ibrahim–Masmoudi–Nakanishi.     

Conservation Laws for Fourth Order Systems in Four Dimensions

In this paper a one-dimensional Keller–Segel model with a logarithmic chemotactic-sensitivity and a non-diffusing chemical is classified with respect to its long time behavior. The strength of production of the non-diffusive chemical has a strong influence on the qualitative behavior of the system concerning existence of global solutions or Dirac-mass formation. Further, the initial data play a crucial role.     

一类四阶非线性抛物型发展方程的吸引子[英文]

本文讨论实轴上的与Cahn-Hilliard方程有联系的一类四阶非线性抛物型方程Эu/Эt σЭ^4u/Эx^4 αu uЭu/Эx g(u)=f(x,t)的长时间行为。在外力f(x,t)是时间t的拟周期函数，非线性项g(u)满足一定的条件下，通过引入过程的概念，证明系统存在一致吸引子，并给出吸引子维数的上界估计。     

一类四阶偏微分多智能体系统的一致性控制

研究一类四阶偏微分多智能体系统的一致性控制问题,该类系统中的每个智能体是由四阶偏微分方程构建而成.针对系统的特点,通过构建合适空间上的Lyapunov泛函,得到分布式反馈控制律.当该反馈控制律作用于系统时,系统状态变量的一致性误差于L2(0,l)×L2(0,l)空间内收敛到零.最后,通过仿真算例验证了方法的有效性.     

四阶障碍问题的稳定化混合有限元方法

本文讨论了四阶障碍问题的稳定化混合有限元方法.首先,引入网格依赖范数,通过加罚方法得到了与四阶障碍问题的等价的混合变分形式.随后给出了基于C~0协调有限元空间(W_h,V_h)的混合有限元逼近,例如P_k-P_k三角形有限元.在网格依赖范数下,(W_h,V_h)满足离散的inf-sup条件.最后,我们在不同的假设下,得到了一些误差估计.     

On the Correctness of the Dirichlet Problem in a Characteristic Rectangle for Fourth Order Linear Hyperbolic Equations

It is proved that the Dirichlet problem is correct in the characteristic rectangle D _ab = [0, a] × [0, b] for the linear hyperbolic equation

On the Dirichlet Problem in a Characteristic Rectangle for Fourth Order Linear Singular Hyperbolic Equations

In the rectangle D = (0,

四阶具强阻尼非线性波动方程解的整体存在性与不存在性

研究四阶具强阻尼非线性波动方程utt-△u+△~2u-α△ut=f(u)的初边值问题.利用位势井方法,得到了问题整体解存在性与不存在性的最佳条件(门槛结果).     

Gradient Flow for the Helfrich Functional

The author studies the L ² gradient flow of the Helfrich functional, which is a functional describing the shapes of human red blood cells. For any ?? _i ?? 0 and c ₀, the author obtains a lower bound on the lifespan of the smooth solution, which depends only on the concentration of curvature for the initial surface.     

Decoupled Mixed Element Methods for Fourth Order Elliptic Optimal Control Problems with Control Constraints

In this paper, we study the finite element methods for distributed optimal control problems governed by the biharmonic operator. Motivated from reducing the regularity of solution space, we use the decoupled mixed element method which was used to approximate the solution of biharmonic equation to solve the fourth order optimal control problems. Two finite element schemes, i.e., Lagrange conforming element combined with full control discretization and the nonconforming Crouzeix-Raviart element combined with variational control discretization, are used to discretize the decoupled optimal control system. The corresponding a priori error estimates are derived under appropriate norms which are then verified by extensive numerical experiments.     

One-Dimensional Symmetry and Liouville Type Results for the Fourth Order Allen-Cahn Equation in \RN

In this paper, the authors prove an analogue of Gibbons' conjecture for the extended fourth order Allen-Cahn equation in R~N, as well as Liouville type results for some solutions converging to the same value at infinity in a given direction. The authors also prove a priori bounds and further one-dimensional symmetry and rigidity results for semilinear fourth order elliptic equations with more general nonlinearities.     

四阶非线性常微分方程非线性三点边值问题解的存在性与唯一性

该文利用“Matching”技巧 ,给出了四阶非线性常微分方程y(4) =f(x,y,y′,y″,y ) ,满足非线性三点边界条件k(y(b) ,y′(b) ,y″(b) ,y (b) ,y(a) ,y′(a) ,y″(a) ,y (a) ) =0 ,y(b) =μg(y′(b) ,y (b) ) =0 ,h(y(b) ,y′(b) ,y″(b) ,y (b) ,y(c) ,y′(c) ,y″(c) ,y (c) ) =0的三点边值问题存在解与存在唯一解的具体的充分条件     

Attractors and Dimensions for Discretizations of a NLS Equation with a Non-local Nonlinear Term

In this paper we consider a semi-dicretized nonlinear Schrödinger (NLS) equation with local integral nonlinearity. It is proved that for each mesh size, there exist attractors for the discretized system. The bounds for the Hausdorff and fractal dimensions of the discrete attractors are obtained, and the various bounds are independent of the mesh sizes. Furthermore, numerical experiments are given and many interesting phenomena are observed such as limit cycles, chaotic attractors and a so-called crisis of the chaotic attractors.     

Inverse Scattering for a Schroedinger Operator with a Repulsive Potential

We consider a pair of Hamiltonians （H, H0） on L2（R^n）, where H0=p^2 -x^2 is a SchrSdinger operator with a repulsive potential, and H = H0＋V（x）. We show that, under suitable assumptions on the decay of the electric potential, V is uniquely determined by the high energy limit of the scattering operator.     

关于非线性Davey-Stewartson方程的散射理论

本文讨论具有三个非线性项的Davey-Stewartson方程的散射算子在整个能量空间H~1中存在。