Vladimir Georgiev, Boris Shakarov
Local and global well - posedness of the solution to the two space dimensional Dirac equation with Hartree type nonlinearity is established with the initial datum in the space $H^s(\mathbb{R}^2, \mathbb{C}^2)$ with $s >0.$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
On Dirac equations with Hartree type nonlinearity in modulation spaces
针对解耦Dirac-Klein-Gordon系统得到的带Hartree型非线性项的Dirac方程，本文通过扩展初始数据函数空间，获得了局部适定性结果，可处理先前未涉及的初始数据。
We study the Cauchy problems for the Hartree-type nonlinear Dirac equations with Yukawa-type potential derived from pseudoscalar field. We establish scattering for large data but with a relatively small part of initial data associated with charge conjugation by exploiting null structure induced by chirality.
We consider the Dirac equation with cubic Hartree-type nonlinearity derived by uncoupling the Dirac-Klein-Gordon systems. We prove small data scattering result in full subcritical range. Main ingredients of the proof are the localized Strichartz estimates, improved bilinear estimates thanks to null-structure hidden in Dirac operator and $Up,Vp$ function spaces. We apply the projection operator and get a system which of linear part is the Klein-Gordon type. It enables us to exploit the null-structures in equation. This result is shown to be almost optimal by showing that iteration method based …
We study the Cauchy problems for the Hartree-type nonlinear Dirac equations with Yukawa-type potential in two and three spatial dimensions. This paper improves our previous results \cite{chohlee,cholee}; we establish global well-posedness and scattering for large data with a certain condition. Firstly we investigate the long-time behavior of solutions to the Dirac equation satisfies good control provided that a particular dispersive norm of solutions is bounded. The key of our proof relies on modifying multilinear estimates obtained in our previous papers. Secondly, we obtain large data scatte…
Recently low-regularity behaviour of solutions to cubic Dirac equations with the Hartree-type nonlinearity has been extensively studied in somewhat a specific assumption on the structure of the nonlinearity. The key approach of previous results was to exploit the null structure in the nonlinearity and the decay of the Yukawa potential. In this paper, we aim to go beyond; we investigate the strong scattering property of cubic Dirac equations with quite a general class of the Hartree-type nonlinearity, which covers the Coulomb potential as well as the Yukawa potential, and the bilinear form, in …
We consider Cauchy problem of the Hartree-type nonlinear Dirac equation with potentials given by $V_b(x) = \frac1{4π}\frac{e^{-b|x|}}{|x|}\, (b \ge 0)$. In previous works, a standard argument is to utilise null form estimates in order to prove global well-posedness for $H^s$-data, $s>0$. However, the null structure inside the equations is not enough to attain the critical regularity. We impose an extra regularity assumption with respect to the angular variable. Firstly, we prove global well-posedness and scattering of Dirac equations with Hartree-type nonlinearity for $b>0$ for small $L^2_x$-d…
In this paper, we study the Cauchy problem of 2d Dirac equation with Hartree type nonlinearity $c(|\cdot|^{-γ} * \langle ψ, βψ\rangle)βψ$ with $c\in \mathbb R\setminus\{0\} $, $0 < γ< 2$. Our aim is to show the small data global well-posedness and scattering in $H^s$ for $s > γ-1$ and $1 < γ< 2$. The difficulty stems from the singularity of the low-frequency part $|ξ|^{-(2-γ)}χ_{\{|ξ|\le 1\}}$ of potential. To overcome it we adapt $U^p-V^p$ space argument and bilinear estimates of \cite{yang, tes2d} arising from the null structure. We also provide nonexistence result for scattering in the long…
We prove that the initial value problem for the Dirac equation $ \left ( -iγ^μ\partial_μ+ m \right) ψ = \left(\frac{e^{- |x|}}{|x|} \ast ( \overline ψψ)\right) ψ\quad \text{in } \ \R^{1+3} $ is globally well-posed and the solution scatters to free waves asymptotically as $t \rightarrow \pm \infty$, if we start with initial data that is small in $H^s$ for $s>0$. This is an almost critical well-posedness result in the sense that $L^2$ is the critical space for the equation. The main ingredients in the proof are Strichartz estimates, space-time bilinear null-form estimates for free waves in $L^2$…
Long-range scattering for 2D Dirac-Hartree equations
该研究针对二维狄拉克-哈特里方程小解，结合时空共振方法与方程零结构，证明其加权索伯列夫空间中小初值的整体适定性与长程散射，克服二维问题时间衰减弱的难点。
We study Cauchy problem for the Klein-Gordon (HNLKG), wave (HNLW) and Schrödinger (HNLS) equations with cubic convolution (Hartree type) nonlinearity. Some global well-posedness and scattering are obtained for the (HNLKG) and (HNLS) with small Cauchy data in some modulation spaces. Global well-posedness for fractional Schrödinger (fNLSH) equation with Hartree type nonlinearity is obtained with Cauchy data in some modulation spaces. Local well-posedness for (HNLW), (fHNLS) and (HNLKG) with rough data in modulation spaces is shown. This improves known results in Sobolev spaces in some sense. As …
We consider the initial value problem for the L^2-critical defocusing Hartree equation in R^n, n \ge 3. We show that the problem is globally well posed in H^s(R^n) when 1>s> \frac{2(n-2)}{3n-4}$. We use the "I-method" combined with a local in time Morawetz estimate for the smoothed out solution.
The aim of this paper is to show the local well-posedness of 2 dimensional Dirac equations with power type and Hartree type nonlinearity derived from honeycomb structure in $H^s$ for $s>\frac78$ and $s>\frac38$, respectively. We also provide the smoothness failure of flows of Dirac equations.
In this paper we consider the defocusing Hartree nonlinear Schrödinger equations on $\mathbb T^3$ with real valued and even potential $V$ and Fourier multiplier decaying like $|k|^{-β}$. By relying on the method of random averaging operators in arXiv:1910.08492, we show that there exists $\frac{1}{2} \ll β_0 <1 $ such that for $ β> β_0 $ we have invariance of the associated Gibbs measure and global existence of strong solutions in its statistical ensemble. In this way we extend Bourgain's seminal result [7] which requires $β>2$ in this case.
We extend the recent work of Chong et al., (2022) to the critical case. More precisely, we prove global in time, uniform in $N$ estimates for the solutions $ϕ$, $Λ$ and $Γ$ of a coupled system of Hartree--Fock--Bogoliubov type with interaction potential $\frac1NV_N(x-y)=N^{2}v(N(x-y))$. We assume that the potential $v$ is small which satisfies some technical conditions, and the initial conditions have finite energy. The main ingredient is a sharp estimate for the linear Schrödinger equation with potential in 6+1 dimension, which may be of interest in its own right.
By $I$-method, the interaction Morawetz estimate, long time Strichartz estimate and local smoothing effect of Schrödinger operator, we show global well-posedness and scattering for the defocusing Hartree equation $$\left\{ \begin{array}{ll} iu_t + Δu &=F(u), \quad (t,x) \in \mathbb{R} \times \mathbb{R}^4 u(0) \\ &=u_0(x)\in H^s(\mathbb{R}^4), \end{array} \right. $$ where $F(u)= (V* |u|^2) u$, and $V(x)=|x|^{-γ}$, $3< γ<4$, with radial data in $H^{s}(\mathbb{R}^4)$ for $s>s_c:=γ/2-1$. It is a sharp global result except of the critical case $s=s_c$, which is a very difficult open problem.
A Scattering Result Around a Non-Localised Equilibria for the Quintic Hartree Equation for Random Fields
本文研究随机场五次哈特里方程在高维情形下非局域平衡态附近的散射行为，将已有的三次哈特里方程散射结果推广到三体相互作用的五次情形，并适用于包含 Dirac delta 在内的广泛势类。
The Global well-posedness for Klein-Gordon-Hartree equation in modulation spaces
该研究利用Bourgain高低频分解方法，首次证明了3维Klein-Gordon-Hartree方程在调制空间中具大初值时的全局适定性，拓展了该方程低正则性问题的研究范畴。
We extend the results of the 2019 paper by the third and fourth author globally in time. More precisely, we prove uniform in $N$ estimates for the solutions $ϕ$, $Λ$ and $Γ$ of a coupled system of Hartree-Fock-Bogoliubov type with interaction potential $V_N(x-y)=N^{3 β}v(N^β(x-y))$ with $β<1$. The potential satisfies some technical conditions, but is not small. The initial conditions have finite energy and the "pair correlation" part satisfies a smallness condition, but are otherwise general functions in suitable Sobolev spaces, and the expected correlations in $Λ$ develop dynamically in time.…
Global well-posedness and scattering for mass-critical Hartree equation
本文在标度临界正则性下证明了质量临界 Hartree 方程的散射猜想，去除了径向限制，涵盖散焦与聚焦（质量小于基态）情形。
Global well-posedness of the energy-critical stochastic Hartree nonlinear wave equation
本文研究了能量临界随机Hartree非线性波方程的整体适定性，利用概率扰动等方法，证明了在能量空间中具有确定性初始数据时整体适定，并在能量空间以下具有随机化初始数据时几乎必然整体适定。
We study the Cauchy problem for fractional Schrödinger equation with cubic convolution nonlinearity ($i\partial_t u - (-Δ)^{\fracα{2}}u\pm (K\ast |u|^2) u =0$) with Cauchy data in the modulation spaces $M^{p,q}(\mathbb R^{d}).$ For $K(x)= |x|^{-γ}$ $ (0< γ< \text{min} \{α, d/2\})$, we establish global well-posedness results in $M^{p,q}(\mathbb R^{d}) (1\leq p \leq 2, 1\leq q < 2d/ (d+γ))$ when $α=2, d\geq 1$, and with radial Cauchy data when $d\geq 2, \frac{2d}{2d-1}< α< 2. $ Similar results are proven in Fourier algebra $\mathcal{F}L^1(\mathbb R^d) \cap L^2(\mathbb R^d).$
Global well-posedness for generalized fractional Hartree equations with rough initial data in all dimensions
本文证明了所有维度下具有粗糙初值的广义分数阶 Hartree 方程在特定调制空间中的整体适定性，核心方法是结合分裂技巧与半群的多项式增长估计。
We prove that certain Sobolev-type norms, slightly stronger than those given by energy conservation, stay bounded uniformly in time and $N$. This allows one to extend the local existence results of the second and third author globally in time. The proof is based on interaction Morawetz-type estimates and Strichartz estimates (including some new end-point results) for the equation $\{ \frac{1}{i}\partial_t-Δ_{x}-Δ_{y}+\frac{1}{N}V_N(x-y) \}Λ(t, x, y) =F$ in mixed coordinates such as $L^p(dt) L^q(dx) L^2(dy)$, $L^p(dt) L^q(dy) L^2(dx)$, $L^p(dt) L^q(d(x-y)) L^2(d(x+y))$. The main new technical i…
This note studies the asymptotic behavior of global solutions to the fourth-order generalized Hartree equation $$i\dot u+Δ^2 u\pm(I_α*|u|^p)|u|^{p-2}u=0.$$ Indeed, for both attractive and repulsive sign, the scattering is obtained in the mass super-critical and energy sub-critical regimes, with radial setting.
This paper studies the asymptotic behavior of global solutions to the generalized Hartree equation $$i\dot u+Δu+(I_α*|\cdot|^b|u|^p)|x|^b|u|^{p-2}u=0 .$$ Indeed, using a new approach due to \cite{dm}, one proves the scattering of the above inhomogeneous Choquard equation in the mass-super-critical and energy sub-critical regimes with radial setting.
This manuscript proves the energy scattering of global solutions to a repulsive fourth-order generalized Hartree equation with non-radial data in the inter-critical regime. This work uses a new approach due to Dodson-Murphy [4] and extends the previous work [14] by removing the spherically symmetric assumption on the data.
Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case
我们建立了能量和$\dot{H}^1$范数小于$\mathbb{R}\times \mathbb{R}^d$，$d\geq 5$中基态能量和$\dot{H}^1$范数的能量临界聚焦Hartree方程径向解的整体存在性和散射性.
In this paper, we consider the Maxwell-Dirac system in 3 dimension under zero magnetic field. We prove the global well-posedness and modified scattering for small solutions in the weighted Sobolev class. Imposing the Lorenz gauge condition, (and taking the Dirac projection operator), it becomes a system of Dirac equations with Hartree type nonlinearity with a long range potential as $|x|^{-1} $. We perform the weighted energy estimates. In this procedure, we have to deal with various resonance functions that stem from the Dirac projections. We use the spacetime resonance argument of Germain-Ma…
We consider the Cauchy problem associated to the fourth-order nonlinear Schrödinger-Hartree equation with variable dispersion coefficients. The variable dispersion coefficients are assumed to be continuous or periodic and piecewise constant in time functions. We prove local and global well-posedness results for initial data in $H^s$-spaces. We also analyze the scaling limit of fast dispersion management and the convergence to a model with averaged dispersions.
We study the global behavior of solutions to the nonlinear generalized Hartree equation, where the nonlinearity is of the non-local type and is expressed as a convolution, $$ i u_t + Δu + (|x|^{-(N-γ)} \ast |u|^p)|u|^{p-2}u=0, \quad x \in \mathbb{R}^N, t\in \mathbb{R}. $$ Our main goal is to understand behavior of $H^1$ (finite energy) solutions of this equation in various settings. In this work we make an initial attempt towards this goal. We first investigate the $H^1$ local wellposedness and small data theory. We then, in the intercritical regime ($0<s<1$), classify the behavior of $H^1$ so…
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.
This paper studies the inhomogeneous fractional Schödinger equation $$i\dot u-(-Δ)^s u=\pm(I_α*|\cdot|^b|u|^p)|x|^b|u|^{p-2}u.$$ In the mass super-critical and energy sub-critical regimes, using a Gagliardo-Nirenberg adapted to the above problem, the standing waves give a sharp threshold of global existence versus finite time blow-up of solutions.
We consider a many-body Boson system with pairwise particle interaction given by $N^{3β-1}v(N^βx)$ for $0<β<1$ and $v$ a non-negative spherically-symmetric function. Our main result is the extension of the local-in-time Fock space (norm) approximation of the exact dynamics of squeezed states proved in Grillakis and Machedon, Comm. PDEs, (2017) for $0<β<\frac{2}{3}$ to a global-in-time norm approximation for $0<β<1$. Our work can also be viewed as a generalization of the results of Boccato, Cenatiempo, and Schlein, Ann. Henri Poincaré, (2017) to a more general set of initial data that includes …
The system describing a single Dirac electron field coupled with classically moving point nuclei is presented and studied. The model is a semi-relativistic extension of corresponding time-dependent one-body Hartree-Fock equation coupled with classical nuclear dynamics, already known and studied both in quantum chemistry and in rigorous mathematical literature. We prove local existence of solutions for data in $H^s$ with $s>1$ and local well posedness in $H^s$ for $s>3/2$. In the course of the analysis a second new result of independent interest is discussed and proved, namely the construction …
We study the generalized Hartree equation, which is a nonlinear Schrödinger-type equation with a nonlocal potential $iu_t + Δu + (|x|^{-b} \ast |u|^p)|u|^{p-2}u=0, x \in \mathbb{R}^N$.We establish the local well-posedness at the non-conserved critical regularity $\dot{H}^{s_c}$ for $s_c \geq 0$, which also includes the energy-supercritical regime $s_c>1$ (thus, complementing the work in [3], where the authors obtained the $H^1$ well-posedness in the intercritical regime together with classification of solutions under the mass-energy threshold). We next extend the local theory to global: for sm…
Global well-posedness and scattering for the mass-critical Hartree equation with radial data
我们建立全球适定性和散射的解决方案的质量临界非线性Hartree方程$iu_t+Δu=\pm(|x|^{-2}*|u|^2)u$大球对称$L^2_x(\Bbb{R}^d)$初始数据，在聚焦的情况下，我们需要，当然，质量严格小于的基态。
Strichartz estimates for the 2D and 3D massless Dirac-Coulomb equations and applications
本文利用既有工具证明了二维和三维无质量 Dirac-Coulomb 方程带有附加角正则性的 Strichartz 估计，并以此展示了带 Hartree 型非线性项的 Dirac-Coulomb 方程的局部适定性结果。
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 global well-posedness for 3D Dirac equation with a concentrated nonlinearity.
Classification of minimal-mass blowup solutions to the mass-critical Hartree equation at the $L^2$-critical level
该文对质量临界Hartree方程在L²临界水平下的最小质量爆破解进行分类，证明全局最小质量非散射解为孤波、有限时间最小质量爆破解为孤波的伪共形像，推广了相关结果并消除了径向假设。
This article is concerned with the mathematical analysis of a class of a nonlinear fractional Schrodinger equations with a general Hartree-type integrand. We prove existence and uniqueness of global-in-time solutions to the associated Cauchy problem. Under suitable assumptions, we also prove the existence of standing waves using the method of concentration-compactness by studying the associated constrained minimization problem. Finally we show the orbital stability of standing waves which are the minimizers of the associate variational problem.
We consider the mass-subcritical Hartree equation with a homogeneous kernel, in the space of square integrable functions whose Fourier transform is integrable. We prove a global well-posedness result in this space. On the other hand, we show that the Cauchy problem is not even locally well-posed if we simply work in the space of functions whose Fourier transform is integrable. Similar results are proven when the kernel is not homogeneous, and is such that its Fourier transform belongs to some Lebesgue space.
Time global wellposedness in L^p for the Chern-Simons-Dirac equation in 1+1 dimension is discussed.
Modified Wave operators for the Hartree equation with repulsive Coulomb potential
本文研究带排斥库仑势的 Hartree 方程的终态问题，将已有工作推广至 Hartree 非线性，构造唯一全局解散射到给定渐近轮廓，并证明修正波算子的存在性。
Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data
我们考虑散焦，$\dot{H}^1$临界Hartree方程的径向数据在所有维度$(n\geq 5)$。我们在能量空间中证明了该模型的全局适定性和散射结果。本文的创新之处在于，我们首先利用定域Morawetz恒等式中的$\displaystyle - \int_{I}\int_{|x|\leq A|I|^{1/2}}|u|^{2}Δ\Big(\frac{1}{|x|}\Big)dxdt$项来排除能量集中的可能性，而不是依赖于非线性的经典Morawetz估计.
We consider the fractional Hartree equation in the $L^2$-supercritical case, and we find a sharp threshold of the scattering versus blow-up dichotomy for radial data: If $ M[u_{0}]^{\frac{s-s_c}{s_c}}E[u_{0}<M[Q]^{\frac{s-s_c}{s_c}}E[Q]$ and $M[u_{0}]^{\frac{s-s_c}{s_c}}\|u_{0}\|^2_{\dot H^s}<M[Q]^{\frac{s-s_c}{s_c}}\| Q\|^2_{\dot H^s}$, then the solution $u(t)$ is globally well-posed and scatters; if $ M[u_{0}]^{\frac{s-s_c}{s_c}}E[u_{0}]<M[Q]^{\frac{s-s_c}{s_c}}E[Q]$ and $M[u_{0}]^{\frac{s-s_c}{s_c}}\|u_{0}\|^2_{\dot H^s}>M[Q]^{\frac{s-s_c}{s_c}}\| Q\|^2_{\dot H^s}$, the solution $u(t)$ blow…
In this article, we prove the global well-posedness of the time-dependent Hartree-Fock-Bogoliubov (TDHFB) equations in $\mathbb{R}^{1+1}$ with two-body interaction potentials of the form $N^{-1}v_N(x) = N^{β-1} v(N^βx)$ where $v$ is a sufficiently regular radial function $v \in L^1(\mathbb{R})\cap C^\infty(\mathbb{R})$. In particular, using methods of dispersive PDEs similar to the ones used in Grillakis and Machedon, Comm. PDEs., (2017), we are able to show for any scaling parameter $β>0$ the TDHFB equations are globally well-posed in some Strichartz-type spaces independent of $N$, cf. (Bach …
We solve globally a radial cubic Dirac equation perturbed with a small potential, with data of small critical norm $H^{1}$. The main tool are new endpoint estimates of the perturbed Dirac flow for a class of radial-type initial data.
We study the Cauchy problem for the fractional Schrödinger equation $$ i\partial_tu = (m^2-Δ)^\frac\alpha2 u + F(u) in \mathbb{R}^{1+n}, $$ where $ n \ge 1$, $m \ge 0$, $1 < α< 2$, and $F$ stands for the nonlinearity of Hartree type: $$F(u) = λ(\frac{ψ(\cdot)}{|\cdot|^γ} * |u|^2)u$$ with $λ= \pm1, 0 <γ< n$, and $0 \le ψ\in L^\infty(\mathbb R^n)$. We prove the existence and uniqueness of local and global solutions for certain $α$, $γ$, $λ$, $ψ$. We also remark on finite time blowup of solutions when $λ= -1$.
We consider the Cauchy problem of massless Dirac-Maxwell equations on an asymptotically flat background and give a global existence and uniqueness theorem for initial values small in an appropriate weighted Sobolev space. The result can be extended via analogous methods to Dirac-Higgs-Yang-Mills theories.
In the article we obtain almost global existence for Dirac Equations with high regularity and small initial datum on Tori. Besides, the global existence with low regularity and small initial datum is gotten. The approaches are mainly Gagliardo-Nirenberg-Moser estimates and Bernstein-Type Lemma.
Global Existence and Time Decay for the Vlasov-Hartree System
研究弗拉索夫 - 哈特里系统的适定性和色散性质，用低正则泛函框架证明其全局适定性，粒子轨迹有意义，初始费米子密度正则时解是经典的，还给出相互作用为斥力和引力时系统的时间衰减及场增长情况。
Construction of a Gibbs measure for the zonal Dirac equation
本文提出了一种构造球面上狄拉克方程吉布斯测度的框架，并通过紧性论证证明了该测度的存在性。
Global-in-time Semiclassical Regularity for the Hartree-Fock Equation
本文证明了在奇异相互作用下，Hartree-Fock 方程的解在任意长时间内保持关于半经典参数的一致正则性，并将从多体动力学推导 Hartree-Fock 和 Vlasov 方程的结论推广到任意长时间。
Global well-posedness of the nonlinear Hartree equation for infinitely many particles with singular interaction
本文研究非线性Hartree方程在奇异相互作用下的全局解的存在性，证明了在有限测度和有限相对熵条件下解的全局存在性。
In this paper, we settle the problem for time-dependent Hartree equation with inhomogeneous boundary condition in a bounded Lipschitz domain in $\mathbb{R}^{N}$. A global existence result is derived.
In this paper, we consider the Cauchy problem for Klein-Gordon equation with a cubic convolution nonlinearity in $\R^3$. By making use of Bourgain's method in conjunction with a precise Strichartz estimate of S.Klainerman and D.Tataru, we establish the $H^s (s<1)$ global well-posedness of the Cauchy problem for the cubic convolution defocusing Klein-Gordon-Hartree equation. Before arriving at the previously discussed conclusion, we obtain global solution for this non-scaling equation with small initial data in $H^{s_0}\times H^{s_0-1}$ where $s_0=\fracγ6$ but not $\frac\gamma2-1$, for this equ…
Global Semiclassical Limit from Hartree to Vlasov Equation for Concentrated Initial Data
本文在维度 d≥3 的奇异势（含三维库仑势）情形下，针对集中初值证明了从Hartree方程到Vlasov方程的定量全局半经典极限，并给出空间与速度矩的半经典界及空间密度渐近行为。
We study a class of two-dimensional non-linear Schrödinger equations with point-like singular perturbation and Hartree non-linearity. The point-like singular perturbation of the free Laplacian induces appropriate perturbed Sobolev spaces that are necessary for the study of ground states and evolution flow. We include in our treatment both mass sub-critical and mass critical Hartree non-linearities. Our analysis is two-fold: we establish existence, symmetry, and regularity of ground states, and we demonstrate the well-posedness of the associated Cauchy problem in the singular perturbed energy s…
Global well-posedness and scattering for the defocusing $H^{\frac12}$-subcritical Hartree equation in $\mathbb{R}^d$
在$\mathbb{R}^d$，$d\geq 3$中证明了具有低正则性数据的散焦$H^{\frac12}$ -亚临界（即$2<γ<3$）Hartree方程的整体适定性和散射性.准确地说，我们表明，一个唯一的和全球性的解决方案存在的Sobolev空间$H^s\big(\mathbb{R}^d\big)$与$s>4(γ-2)/(3γ-4)$，这也分散在两个时间方向上的初始数据。这改进了\cite{ChHKY}中的结果，其中为任何$s>\max\big(1/2,4(γ-2)/(3γ-4)\big)$建立了全局适定性。在我们的证明中的新成分是，我们使用的相互作用Morawetz估计的平滑的解决方案$Iu$，而不是一个相互作用Morawetz估计的解决方案$u$，我们仔细分析的单调性性质的乘数$m(ξ)\cdot < ξ>^p$。作为我们的证明的副产品，我们得到的$H^s$规范的解决方案服从统一的时间界限。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。