证明了描述吸引费米子系统演化的Hartree-Fock和Hartree-Fock-Bogoliubov型方程的球对称解和负能量解的有限时间爆破。G.白矮星）。我们的主要结果有两个方面：首先，我们将[Hainzl and Schlein，Comm. Math. Phys. \textbf{287}（2009），705- 714]的最近的爆破结果推广到具有无限秩解和一般类牛顿型相互作用的Hartree-Fock方程。其次，我们证明了球对称的Hartree-Fock-Bogoliubov模型的解的有限时间爆破的存在性，其中引入了角动量截断。我们还解释了在整个Hartree-Fock-Bogoliubov理论中遇到的关键困难。
英文摘要：
We prove finite-time blowup for spherically symmetric and negative energy solutions of Hartree-Fock and Hartree-Fock-Bogoliubov type equations, which describe the evolution of attractive fermionic systems (e. g. white dwarfs). Our main results are twofold: First, we extend the recent blowup result of [Hainzl and Schlein, Comm. Math. Phys. \textbf{287} (2009), 705--714] to Hartree-Fock equations with infinite rank solutions and a general class of Newtonian type interactions. Second, we show the existence of finite-time blowup for spherically symmetric solutions of a Hartree-Fock-Bogoliubov model, where an angular momentum cutoff is introduced. We also explain the key difficulties encountered in the full Hartree-Fock-Bogoliubov theory.
We prove decay with respect to some Lebesgue norms for a class of Schrödinger equations with non-local nonlinearities by showing new Morawetz inequalities and estimates. As a byproduct, we obtain large-data scattering in the energy space for the solutions to the systems of $N$ defocusing Schrödinger-Choquard equations with mass-energy intercritical nonlinearities in any space dimension and of defocusing Hartree-Fock equations, for any dimension $d\geq3$.
We demonstrate that the reduced Hartree-Fock equation (REHF) with a small Anderson type background charge distribution has an unique stationary solution by explicitly computing a screening mass at positive temperature.
We prove analytic-type estimates in weighted Sobolev spaces on the eigenfunctions of a class of elliptic and nonlinear eigenvalue problems with singular potentials, which includes the Hartree-Fock equations. Going beyond classical results on the analyticity of the wavefunctions away from the nuclei, we prove weighted estimates locally at each singular point, with precise control of the derivatives of all orders. Our estimates have far-reaching consequences for the approximation of the eigenfunctions of the problems considered, and they can be used to prove a priori estimates on the numerical s…
We show that there is an one-to-one correspondence between solutions to the Poisson-Landscape equations and the reduced Hartree-Fock equations in the semi-classical limit at low temperature. Moreover, we prove that the difference between the two corresponding solutions is small by providing explicit estimates.
Classification of minimal-mass blowup solutions to the mass-critical Hartree equation at the $L^2$-critical level
Graduate School of China Academy of Engineering Physics（中国工程物理研究院研究生院）；School of Mathematics and Physics, University of Science and Technology Beijing（北京科技大学数理学院）；Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）
We consider the gravitational collapse for neutron stars in the Hartree-Fock-Bogoliubov theory. We prove that when the number particle becomes large and the gravitational constant is small such that the attractive interaction strength approaches the Chandrasekhar limit mass slowly, the minimizers develop a universal blow-up profile. It is given by the Lane-Emden solution.
We establish some local and global well-posedness for Hartree-Fock equations of $N$ particles (HFP) with Cauchy data in Lebesgue spaces $L^p \cap L^2 $ for $1\leq p \leq \infty$. Similar results are proven for fractional HFP in Fourier-Lebesgue spaces $ \hat{L}^p \cap L^2 \ (1\leq p \leq \infty).$ On the other hand, we show that the Cauchy problem for HFP is ill-posed if we simply work in $\hat{L}^p \ (2<p\leq \infty).$ Analogue results hold for reduced HFP. In the process, we prove the boundedeness of various trilinear estimates for Hartree type non linearity in these spaces which may be of i…
We prove the decay in the energy space for the solution to the defocusing biharmonic Hartree-Fock equations with mass-supercritical and energy-subcritical Choquard-type nonlinearity in space dimension $d\geq3$. We treat both the free and the perturbed by a potential case. As a direct consequence, we obtain large-data scattering in $H^2(\mathbb R^d)^N, \, N\geq 1$.
We establish both a local and a global well-posedness theories for the nonlinear Hartree-Fock equations and its reduced analog in the setting of the modulation spaces on $\mathbb R^d$. In addition, we prove similar results when a harmonic potential is added to the equations. In the process, we prove the boundedeness of certain multilinear operators on products of the modulation spaces which may be of independent interest.
In this paper we consider an Hartree-Fock type system made by two Schrödinger equations in presence of a Coulomb interacting term and a cooperative pure power and subcritical nonlinearity, driven by a suitable parameter $β\geq 0$. We show the existence of semitrivial and vectorial ground states solutions depending on the parameters involved. The asymptotic behavior with respect to the parameter $β$ of these solutions is also studied.
