Quantum Gravitational Effects on the Boundary

Quantum gravitational effects might hold the key to some of the outstanding problems in theoretical physics. We analyze the perturbative quantum effects on the boundary of a gravitational system and the Dirichlet boundary condition imposed at the classical level. Our analysis reveals that for a black hole solution, there is a contradiction between the quantum effects and the Dirichlet boundary condition: the black hole solution of the one-particle-irreducible action no longer satisfies the Dirichlet boundary condition as would be expected without going into details. The analysis also suggests that the tension between the Dirichlet boundary condition and loop effects is connected with a certain mechanism of information storage on the boundary.     

Nonlinear Hartree equation in high energy-mass

This paper is concerned with the Cauchy problem of the nonlinear Hartree equation. By constructing a constrained variational problem, we get a refined Gagliardo–Nirenberg inequality and the best constant for this inequality. We thus derive two conclusions. Firstly, by establishing and analyzing the invariant manifolds, we obtain a new criteria for global existence and blowup of the solutions. Secondly, we get other sufficient condition for global existence with the discussing of the Bootstrap argument. And based on these two conclusions, we also deduce so-called energy-mass control maps, which expose the relationship between the initial data and the solutions.     

The defocusing energy-supercritical Hartree equation

In this paper,we study the global well-posedness and scattering problem for the energysupercritical Hartree equation iut+Δu.(|x|.γ.|u|2)u=0 with γ4 in dimension d γ.We prove that if the solution u is apriorily bounded in the critical Sobolev space,that is,u ∈Lt∞(I;Hxsc(Rd)) with sc:= γ/2.11,then u is global and scatters.The impetus to consider this problem stems from a series of recent works for the energy-supercritical nonlinear wave equation(NLW) and nonlinear Schrdinger equation(NLS).We utilize the strategy derived from concentration compactness ideas to show that the proof of the global well-posedness and scattering is reduced to disprove the existence of three scenarios:finite time blowup;soliton-like solution and low to high frequency cascade.Making use of the No-waste Duhamel formula,we deduce that the energy of the finite time blow-up solution is zero and so get a contradiction.Finally,we adopt the double Duhamel trick,the interaction Morawetz estimate and interpolation to kill the last two scenarios.     

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.     

The focusing energy-critical Hartree equation

We consider the focusing energy-critical nonlinear Hartree equation iut+Δu=−(⁻⁴|x|∗²|u|)u. We proved that if a maximal-lifespan solution u:I×Rd→C satisfies supt∈I‖∇u(t)_‖2<‖∇W_‖2, where W is the static solution of the equation, then the maximal-lifespan I=R, moreover, the solution scatters in both time directions. For spherically symmetric initial data, similar result has been obtained in [C. Miao, G. Xu, L. Zhao, Global wellposedness, scattering and blowup for the energy-critical, focusing Hartree equation in the radial case, Colloq. Math., in press]. The argument is an adaptation of the recent work of R. Killip and M. Visan [R. Killip, M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, preprint] on energy-critical nonlinear Schrödinger equations.     

On phase segregation in nonlocal two-particle Hartree systems

We prove the phase segregation phenomenon to occur in the ground state solutions of an interacting system of two self-coupled repulsive Hartree equations for large nonlinear and nonlocal interactions. A self-consistent numerical investigation visualizes the approach to this segregated regime.      

引力规范理论中的一类引力波方程

该文给出了Vierbein表述的局域Lorentz群引力规范理论中的一类引力波方程。证明了Bondi平面波方程和引力孤立波方程均被该类方程所包含,这些方程的解均为该类方程在一定条件下的特解。因而这些解是与量子场论协调一致的。     

Rate of Convergence Towards Semi-Relativistic Hartree Dynamics

We consider the semi-relativistic system of N gravitating Bosons with gravitation constant G. The time evolution of the system is described by the relativistic dispersion law, and we assume the mean-field scaling of the interaction where N → ∞ and G → 0 while GN = λ fixed. In the super-critical regime of large λ, we introduce the regularized interaction where the cutoff vanishes as N → ∞. We show that the difference between the many-body semi-relativistic Schrödinger dynamics and the corresponding semi-relativistic Hartree dynamics is at most of order N ^?1 for all λ, i.e., the result covers the sub-critical regime and the super-critical regime. The N dependence of the bound is optimal.     

具弱耗散项的非线性Hartree方程

该文讨论了一类在量子理论中有着较多应用的具耗散项的非线性Hartree方程。分别从耗散系数和初值两个方面讨论了解的整体存在性条件。一方面,利用Strichartz估计,得到仅依赖于耗散系数的整体存在性条件。另一方面,也得到了仅依赖于初值大小的整体存在性条件。而且,还得到了一个整体解存在的小初值准则。     

Gravitational experiments in space

Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 78, No. 1, pp. 3–10, January, 1989.     

Quasiclassical soliton solutions of the Hartree equation

One considers the three-dimensional Hartree equations , ΔU=¦ψ¦². One constructs the asymptotic (h→0) solution ψ of soliton type, localized mod 0 (h^∞) in a compact domain. One finds the corresponding asymptotics of the self-consistent potential U. One obtains quantification conditions on the energy of the soliton.     

Semi-Classical Analysis for Hartree Equations in Some Supercritical Cases

We study the semi-classical limit of the Hartree equation, which has focusing at a point. There exists a critical index indicating nonlinear effect around the caustic, and it is known that the influence by the nonlinearity is negligible in subcritical case (called linear caustic case), and that it is not in critical case (nonlinear caustic case). We give the asymptotic behavior beyond caustic in some supercritical cases which give rise to very strong nonlinear effect. Submitted: August 25, 2006. Accepted: December 11, 2006.     

Remarks about a generalized pseudo-relativistic Hartree equation

With appropriate hypotheses on the nonlinearity f, we prove the existence of a ground state solution u for the problem

${(?\mathrm{\Delta }+m^2)}^{\sigma }u+Vu=(W?F(u))f(u)\text{in }{\mathbb{R}}^N,$

where $0<\sigma <1$, V is a bounded continuous potential and F the primitive of f. We also show results about the regularity of any solution of this problem.