We investigate the well-posedness in the generalized Hartree equation $iu_t + Δu + (|x|^{-(N-γ)} \ast |u|^p)|u|^{p-2}u=0$, $x \in \mathbb{R}^N$, $0<γ<N$, for low powers of nonlinearity, $p<2$. We establish the local well-posedness for a class of data in weighted Sobolev spaces, following ideas of Cazenave and Naumkin [6]. This crucially relies on the boundedness of the Riesz transform in weighted Lebesgue spaces. As a consequence, we obtain a class of data that exists globally, moreover, scatters in positive time. Furthermore, in the focusing case in the $L^2$-supercritical setting we obtain a subset of locally well-posed data with positive energy, which blows up in finite time.
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…
With appropriate hypotheses on the nonlinearity $f$, we prove the existence of a ground state solution $u$ for the problem \[(-Δ+m^2)^σu+Vu=\left(W*F(u)\right)f(u)\ \ \text{in }\ \mathbb{R}^{N},\] where $0<σ<1$, $V$ is a bounded continuous potential and $F$ the primitive of $f$. We also show results about the regularity of any solution of this problem.
With appropriate hypotheses on the nonlinearity $f$, we prove the existence of a ground state solution $u$ for the problem \[\sqrt{-Δ+m^2}\, u+Vu=\left(W*F(u)\right)f(u)\ \ \text{in }\ \mathbb{R}^{N},\] where $V$ is a bounded potential, not necessarily continuous, and $F$ the primitive of $f$. We also show that any of this problem is a classical solution. Furthermore, we prove that the ground state solution has exponential decay.
We study the global behavior of solutions to the focusing generalized Hartree equation with $H^1$ data at mass-energy threshold in the inter-range case. In the earlier works of Arora-Roudenko [Arora-Roudenko 2021], the behavior of solutions below the mass-energy threshold was classified. In this paper, we first exhibit three special solutions: $e^{it} Q$, $Q^\pm$, where $Q^\pm$ exponentially approach to the $e^{it} Q$ in the positive time direction, $Q^+$ blows up and $Q^-$ scatters in the negative time direction. Then we classify solutions at this threshold, showing that they behave exactly a…
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…
We consider the focusing nonlinear Schrödinger equation $i u_t + Δu + |u|^{p-1}u=0$, $p>1,$ and the generalized Hartree equation $iv_t + Δv + (|x|^{-(N-γ)}\ast |v|^p)|v|^{p-2}u=0$, $p\geq2$, $γ<N$, in the mass-supercritical and energy-subcritical setting. With the initial data $u_0\in H^1(\mathbb{R}^N)$ the characterization of solutions behavior under the mass-energy threshold is known for the NLS case from the works of Holmer and Roudenko in the radial [16] and Duyckaerts, Holmer and Roudenko in the nonradial setting [10] and further generalizations (see [1,11,14]); for the generalized Hartre…
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.
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 mode…
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 study the asymptotic behavior of solutions to the defocusing mass-subcritical Hartree NLS $iu_t + Δu = F(u) = (|x|^{-γ}*|u|^2)u$ on $\mathbb{R}^d$, $d\geq 2$, $\frac{4}{3} < γ< 2$. We show that the scattering problem associated to this equation is analytically well-posed in the weighted spaces $Σ= H^1\cap\mathcal{F}H^1$ and $\mathcal{F}H^1$. Furthermore, we show that the same problem fails to be analytically well-posed for data in $L^2$. This constitutes an infinite loss of regularity between the scattering problems in weighted spaces and in $L^2$. This further develops an earlier investiga…
A priori estimates for the weak solutions the Dirichlet problem for the uniformly higher-order elliptic equations in a smooth bounded domain $Ω\subset \Rn$ in generalized weighted Sobolev-Morrey spaces are obtained.
The goal of this work is to show that the generalized Korn inequality that replaces the symmetric part of the differential matrix in the classical Korn inequality by its trace-free part is valid over John domains and weighted Sobolev spaces. The weights considered are nonnegative powers of the distance to the boundary.
In this paper, we prove a quantitative version of the semiclassical limit from the Hartree to the Vlasov equation with singular interaction, including the Coulomb potential. To reach this objective, we also prove the propagation of velocity moments and weighted Schatten norms which implies the boundedness of the space density of particles uniformly in the Planck constant.
We study the initial value problem associated to the dispersion generalized Benjamin-Ono equation. Our aim is to establish well posedness results in weighted Sobolev spaces and to deduce from them some sharp unique continuation properties of solutions to this equation. In particular, we shall establish optimal decay rate for the solutions of this model.
We consider the well-posedness of the initial value problem associated to the k-generalized Zakharov-Kuznetsov equation in fractional weighted Sobolev spaces. Our method of proof is based on the contraction mapping principle and it mainly relies on the well-posedness results recently obtained for this equation in the Sobolev spaces H^s(\R^2) and a new pointwise commutator type formula involving the group induced by the linear part of the equation and the fractional anisotropic weights to be considered
We consider a generalized steady Stokes system with discontinuous coefficients in a nonsmooth domain when the inhomogeneous term belongs to a weighted $L^q$ space for $2<q<\infty$. We prove the global weighted $L^q$-estimates for the gradient of the weak solution and an associated pressure under the assumptions that the coefficients have small BMO (bounded mean oscillation) semi-norms and the domain is sufficiently flat in the Reifenberg sense. On the other hand, a given weight is assumed to belong to a Muckenhoupt class. Our result generalizes the global $W^{1.q}$ estimate of Calderón-Zygmund…
In this paper, we explore the relations between different kinds of Strichartz estimates and give new estimates in Euclidean space $\mathbb{R}^n$. In particular, we prove the generalized and weighted Strichartz estimates for a large class of dispersive operators including the Schrödinger and wave equation. As a sample application of these new estimates, we are able to prove the Strauss conjecture with low regularity for dimension 2 and 3.
In this paper we study the initial-value problem associated with the dispersion generalized-Benjamin-Ono-Zakharov-Kuznetsov equation, $$ u_{t}+D^{a+1}_x \partial_{x}u+u_{xyy}+uu_{x}=0, \qquad a\in(0,1). $$ More specifically, we study the persistence property of the solution in the weighted anisotropic Sobolev spaces $$ H^{(1+a)s,2s}(\R^{2})\cap L^{2}((x^{2r_1} +y^{2r_2})dxdy), $$ for appropriate $s$, $r_1$ and $r_2$. By establishing unique continuation properties we also show that our results are sharp with respect to the decay in the $x$-direction.
We study stable blow-up dynamics in the generalized Hartree equation with radial symmetry, a Schrödinger-type equation with a nonlocal, convolution-type nonlinearity: $iu_t+Δu +\left(|x|^{-(d-2)} \ast |u|^{p} \right) |u|^{p-2}u = 0, x \in \mathbb{R}^d$. First, we consider the $L^2$-critical case in dimensions d=3, 4, 5, 6, 7 and obtain that a generic blow-up has a self-similar structure and exhibits not only the square root blowup rate but also the log-log correction (via asymptotic analysis and functional fitting). In this setting we also study blow-up profiles and show that generic blow-up s…
We consider the question of scattering for the boson star equation in three space dimensions. This is a semi-relativistic Klein-Gordon equation with a cubic nonlinearity of Hartree type. We combine weighted estimates, obtained by exploiting a special null structure present in the equation, and a refined asymptotic analysis performed in Fourier space, to obtain global solutions evolving from small and localized Cauchy data. We describe the behavior at infinity of such solutions by identifying a suitable nonlinear asymptotic correction to scattering. As a byproduct of the weighted energy estimat…
In this text we study a generalized conservation property for the heat semigroup generated by a Schrödinger operator with nonnegative potential on a weighted manifold. We establish Khasminskii's criterion for the generalized conservation property and discuss several applications.
A Gauge-Covariant Semiclassical Limit from the magnetic Hartree--Fock equation to the magnetic Vlasov--Poisson equation
Department of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）；Laboratory of Mathematics and Complex Systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院，教育部数学与复杂系统重点实验室）
