Stability of standing waves for the L 2-critical Hartree equations with harmonic potential

This article is concerned with the Hartree equations with harmonic potential. By an elaborate mathematical analysis, we obtain a sharp stability threshold of this equation. Then with this threshold, we prove that the standing wave of this equation exists and is stable.     

Stability for the time-dependent Hartree equation with positive energy

In this paper, we show that the H¹ solutions to the time-dependent Hartree equation

     

On a conjecture of Murty and Simon on diameter 2-critical graphs

A graph G is diameter 2-critical if its diameter is two, and the deletion of any edge increases the diameter. Murty and Simon conjectured that the number of edges in a diameter 2-critical graph of order n is at most n²/4 and that the extremal graphs are complete bipartite graphs with equal size partite sets. We use an association with total domination to prove the conjecture for the graphs whose complements have diameter three.     

非线性双曲型Schr(o)dinger方程L2整体解的存在性

本文讨论含L2次临界指数非线性项的广义Schrodinger方程柯西问题,用Strichartz不等式和压缩映射原理证明了在L2初值条件下方程有整体解,即u(t)∈C(R,L2(Rn)),而且证明了含L2临界指数非线性项的广义Schrodinger方程有小初值L2整体解.     

A maximum degree theorem for diameter-2-critical graphs

A graph is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. Let G be a diameter-2-critical graph of order n. Murty and Simon conjectured that the number of edges in G is at most ?n ²/4? and that the extremal graphs are the complete bipartite graphs K _?n/2?,?n/2?. Fan [Discrete Math. 67 (1987), 235–240] proved the conjecture for n ≤ 24 and for n = 26, while Füredi [J. Graph Theory 16 (1992), 81–98] proved the conjecture for n > n ₀ where n ₀ is a tower of 2’s of height about 10¹⁴. The conjecture has yet to be proven for other values of n. Let Δ denote the maximum degree of G. We prove the following maximum degree theorems for diameter-2-critical graphs. If Δ ≥ 0.7 n, then the Murty-Simon Conjecture is true. If n ≥ 2000 and Δ ≥ 0.6789 n, then the Murty-Simon Conjecture is true.     

On the edge-density of 4-critical graphs

Gallai conjectured that every 4-critical graph on n vertices has at least 5/3n-2/3 edges. We prove this conjecture for 4-critical graphs in which the subgraph induced by vertices of degree 3 is connected.     

Stability of standing waves for NLS with perturbed Lamé potential

We perturb a linear Schrödinger equation with Lamé potential with a small positive or negative potential. The new perturbed operator has one or more eigenvalues, at most one in each spectral gap. We then add a nonlinear term and study the stability of the corresponding nonlinear stationary waves.     

On global existence for mass-supercritical nonlinear fractional Hartree equations

In this paper, we consider the nonlinear fractional Schrödinger equations with Hartree type nonlinearity in mass-supercritical and energy-subcritical case. By sharp Hardy-Littlewood-Sobolev inequality and the Pohozaev identity, we established a threshold condition, which leads to a global existence of solutions in energy space.     

Stability of Blowup for a 1D Model of Axisymmetric 3D Euler Equation

The question of the global regularity versus finite- time blowup in solutions of the 3D incompressible Euler equation is a major open problem of modern applied analysis. In this paper, we study a class of one-dimensional models of the axisymmetric hyperbolic boundary blow-up scenario for the 3D Euler equation proposed by Hou and Luo (Multiscale Model Simul 12:1722–1776, 2014) based on extensive numerical simulations. These models generalize the 1D Hou–Luo model suggested in Hou and Luo Luo and Hou (2014), for which finite-time blowup has been established in Choi et al. (arXiv preprint. arXiv:1407.4776, 2014). The main new aspects of this work are twofold. First, we establish finite-time blowup for a model that is a closer approximation of the three-dimensional case than the original Hou–Luo model, in the sense that it contains relevant lower-order terms in the Biot–Savart law that have been discarded in Hou and Luo Choi et al. (2014). Secondly, we show that the blow-up mechanism is quite robust, by considering a broader family of models with the same main term as in the Hou–Luo model. Such blow-up stability result may be useful in further work on understanding the 3D hyperbolic blow-up scenario.     

Global Existence and Scattering in L2 for the Mass-critical Hartree Equation With H0,1 Initial Data

In this paper we consider the Cauchy Problem for the mass-critical Hartree equation I(e)tu △u=μ(|x|2*|u|2)u,(t,x)∈R×Rn,n≥3,(1) u(0,x)=φ(x), x∈Rn,(2)     

推广的GDGH2系统的自相似解及爆破现象

本文研究了广义两分量Dullin-Gottwald-Holm (GDGH2)浅水波系统及其推广形式的一类自相似解.首先通过构造Emden方程,分析了解的全局存在性,以及在一定条件下解的爆破现象;其次利用扰动方法和特征线法,构造了两种形式的精确解.     

On invariant tori of full dimension for 1D periodic NLS

Consider the NLS with periodic boundary conditions in 1D

(0.1)     

On asymptotic stability in energy space of ground states of NLS in 2D

We transpose work by K. Yajima and by T. Mizumachi to prove dispersive and smoothing estimates for dispersive solutions of the linearization at a ground state of a Nonlinear Schrödinger equation (NLS) in 2D. As an application we extend to dimension 2D a result on asymptotic stability of ground states of NLS proved in the literature for all dimensions different from 2.     

Scattering theory for the Dirac equation of Hartree type and the semirelativistic Hartree equation

Consider a scattering problem for the Dirac equation with a nonlocal term including the Hartree type. We improve the condition of the potential term to show the existence of scattering operators for small initial data in the subcritical Sobolev spaces. Our proofs can be applied to the case of the semirelativistic Hartree equation, and lead to improvement of the condition of the potential.