Hartree Equation Research Articles

Energy scattering for the focusing fractional generalized Hartree equation

<p style='text-indent:20px;'>This note studies the asymptotics of radial global solutions to the non-linear fractional Schrödinger equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ i\dot u-(-\Delta)^s u+|u|^{p-2}(I_\alpha *|u|^p)u = 0. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Indeed, using a new method due to Dodson-Murphy [<xref ref-type="bibr" rid="b10">10</xref>], one proves that, in the inter-critical regime, under the ground state threshold, the radial global solutions scatter in the energy space.</p>

Symmetry of positive solutions for systems of fractional Hartree equations

<p style='text-indent:20px;'>In this paper, we deal with a system of fractional Hartree equations. By means of a direct method of moving planes, the radial symmetry and monotonicity of positive solutions are presented.</p>

Global Behavior of Solutions to the Focusing Generalized Hartree Equation

We study behavior of solutions to the nonlinear generalized Hartree equation, where the nonlinearity is of nonlocal type and is expressed as a convolution iut+Δu+(|x|−(N−γ)∗|u|p)|u|p−2u=0,x∈RN,t∈R. Our main goal is to understand global behavior of solutions of this equation in various settings. In this work we make an initial attempt towards this goal and study H1 (finite energy) solutions. We first investigate the H1 local well-posedness and small data theory. We then, in the intercritical regime (0<s<1), classify the behavior of H1 solutions under the mass-energy assumption ME[u0]<1, identifying the sharp threshold for global versus finite time solutions via the sharp constant of the corresponding convolution type Gagliardo–Nirenberg interpolation inequality (note that the uniqueness of a ground state is not known in the general case). In particular, depending on the size of the initial mass and gradient, solutions will either exist for all time and scatter in H1, or blow up in finite time, or diverge along an infinite time sequence. To obtain H1 scattering or divergence to infinity, in this paper we employ the well-known concentration compactness and rigidity method of Kenig and Merle [36] with the novelty of studying the nonlocal, convolution nonlinearity.

On the focusing generalized Hartree equation

In this paper we give a review of the recent progress on the focusing generalized Hartree equation, which is a nonlinear Schrodinger-type equation with the nonlocal nonlinearity, expressed as a convolution with the Riesz potential. We describe the local well-posedness in H1 and Hs settings, discuss the extension to the global existence and scattering, or finite time blow-up. We point out different techniques used to obtain the above results, and then show the numerical investigations of the stable blow-up in the L2 -critical setting. We finish by showing known analytical results about the stable blow-up dynamics in the L2 -critical setting.

A new proof of scattering for the 5D radial focusing Hartree equation

We revisit the scattering result below the ground state of Y. Gao and H. Wu [Scattering for the focusing -critical Hartree equation in energy space. Nonlinear Anal. 2010;73:1043–1056] on the radial focusing energy-subcritical Hartree equation in d = 5, using the method in Dodson and Murphy [A new proof of scattering below the ground state for the 3D radial focusing cubic NLS. preprint, arXiv:1712.09962]. Using the radial Sobolev embedding and a virial-Morawetz type estimate we can exclude the concentration of mass near the origin instead of using the concentration compactness method in Gao and Wu [Scattering for the focusing -critical Hartree equation in energy space. Nonlinear Anal. 2010;73:1043–1056].

Well-posedness and blow-up properties for the generalized Hartree equation

We study the generalized Hartree equation, which is a nonlinear Schrödinger-type equation with a nonlocal potential [Formula: see text]. We establish the local well-posedness at the nonconserved critical regularity [Formula: see text] for [Formula: see text], which also includes the energy-supercritical regime [Formula: see text] (thus, complementing the work in [A. K. Arora and S. Roudenko, Global behavior of solutions to the focusing generalized Hartree equation, Michigan Math J., forthcoming], where we obtained the [Formula: see text] 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 small data we obtain global in time existence and for initial data with positive energy and certain size of variance we show the finite time blow-up (blow-up criterion). In the intercritical setting the criterion produces blow-up solutions with the initial values above the mass–energy threshold. We conclude with examples showing currently known thresholds for global vs. finite time behavior.

Semi-classical Limit of Quantum Free Energy Minimizers for the Gravitational Hartree Equation

For the gravitational Vlasov-Poisson equation, Guo and Rein constructed a class of classical isotropic states as minimizers of free energies (or energy-Casimir functionals) under mass constraints. For the quantum counterpart, that is, the gravitational Hartree equation, isotropic states are constructed as free energy minimizers by Aki, Dolbeault and Sparber. In this paper, we are concerned with the correspondence between quantum and classical isotropic states. Precisely, we prove that as the Planck constant $\hbar$ goes to zero, free energy minimizers for the Hartree equation converge to those for the Vlasov-Poisson equation in terms of potential functions as well as via the Husimi transform and the T\"oplitz quantization.

Focusing nonlinear Hartree equation with inverse‐square potential

AbstractIn this paper, we consider the scattering theory of the radial solution to focusing energy‐subcritical Hartree equation with inverse‐square potential in the energy space using the method from [4]. The main difficulties are due to the fact that the equation is not space‐translation invariant and that the nonlinearity is non‐local. Using the radial Sobolev embedding and a virial‐Morawetz type estimate we can exclude the concentration of mass near the origin. Besides, we can overcome the weak dispersive estimate when , using the dispersive estimate established by [23].

The Hartree and Vlasov equations at positive density

We consider the nonlinear Hartree and Vlasov equations around a translation-invariant (homogeneous) stationary state in infinite volume, for a short range interaction potential. For both models, we consider time-dependent solutions which have a finite relative energy with respect to the reference translation-invariant state. We prove the convergence of the Hartree solutions to the Vlasov ones in a semi-classical limit and obtain as a by-product global well-posedness of the Vlasov equation in the (relative) energy space.

Nodal solutions for pseudo-relativistic Hartree equations

In this paper, we investigate the existence of nodal solutions for the following problem [Display omitted] where 12<α<1,0<μ≤2 and 2N−μN−1<p<2N−μN−2α. We show that for any positive integer k, problem (P0) possesses a radially symmetrical solution changing sign exactly k-times.

Non-Polar Wurtzite (1120) GaN/AlN Quantum Dots for Highly Efficient Opto-Electronic Devices

In III-nitride quantum dots (QDs), optical transition rate is very low because of the large built-in electrostatic field caused by the spontaneous polarization (SP) and piezoelectric (PZ) effects. In this work, we study the screening potential which is a solution of the self-consistent Hartree equation taking into account the built-in electrostatic field and its effect on light emission characteristics of non-polar wurtzite (WZ) (112¯0) GaN/AlN QD. It is found that the light emission intensity of the non-polar (112¯0) GaN/AlN QD structure is expected to be about four times larger than that of the c-plane (0001) GaN/AlN QD structure because the y-polarized matrix elements in the non-polar QD are larger than that in the c-plane QD. These predictions indicate that non-polar GaN/AlN QD structure have strong potential for highly efficient opto-electronic devices.

Stable blow‐up dynamics in the ‐critical and ‐supercritical generalized Hartree equation

AbstractWe study stable blow‐up dynamics in the generalized Hartree equation with radial symmetry, which is a Schrödinger‐type equation with a nonlocal, convolution‐type nonlinearity: urn:x-wiley:00222526:media:sapm12328:sapm12328-math-0003First, we consider the ‐critical case in dimensions 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), thus, behaving similarly to the stable blow‐up regime in the ‐critical nonlinear Schrödinger equation. In this setting, we also study blow‐up profiles and show that generic blow‐up solutions converge to the rescaled , a ground state solution of the elliptic equation .We also consider the ‐supercritical case in dimensions . We derive the profile equation for the self‐similar blow‐up and establish the existence and local uniqueness of its solutions. As in the NLS ‐supercritical regime, the profile equation exhibits branches of nonoscillating, polynomially decaying (multi‐bump) solutions. A numerical scheme of putting constraints into solving the corresponding ordinary differential equation is applied during the process of finding the multi‐bump solutions. Direct numerical simulation of solutions to the generalized Hartree equation by the dynamic rescaling method indicates that the is the profile for the stable blow‐up. In this supercritical case, we obtain the blow‐up rate without any correction. This blow‐up happens at the focusing level , and thus, numerically observable (unlike the ‐critical case). In summary, we find that the results are similar to the behavior of stable self‐similar blowup solutions in the corresponding settings for the nonlinear Schrödinger equation. Consequently, one may expect that the form of the nonlinearity in the Schrödinger‐type equations is not essential in the stable formation of singularities.

On Well-Posedness for General Hierarchy Equations of Gross–Pitaevskii and Hartree Type

Gross–Pitaevskii and Hartree hierarchies are infinite systems of coupled PDEs emerging naturally from the mean field theory of Bose gases. Their solutions are known to be related to initial value problems, in particular the Gross–Pitaevskii and Hartree equations. Due to their physical and mathematical relevance, the issues of well-posedness and uniqueness for these hierarchies have recently been studied thoroughly using specific nonlinear and combinatorial techniques. In this article, we introduce a new approach for the study of such hierarchy equations by firstly establishing a duality between them and certain Liouville equations, and secondly, solving the uniqueness and existence questions for the latter. As an outcome, we formulate a hierarchy equation starting from any initial value problem which is U(1)-invariant and prove a general principle which can be stated formally as follows: In particular, several new well-posedness results, as well as a counterexample to uniqueness for the Gross–Pitaevskii hierarchy equation, are proved. The novelty in our work lies in the aforementioned duality and the use of Liouville equations with powerful transport techniques extended to infinite dimensional functional spaces.

Asymptotic behavior of ground states of 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 − Δ + m 2 u + V u = ( W ∗ F ( u ) ) f ( u ) in R N , where V is a bounded potential, not necessarily continuous, and F the primit

Blow-Up Profile of Pseudo-relativistic Hartree Equations with Singular Potentials

We consider the minimizers for the nonlinear Schrodinger functional $$\begin{aligned} \mathcal {E}_a(u)=\int _{{\mathbb {R}}^3}\left( u\sqrt{-\Delta +m^2}u+V(x)u^2\right) \hbox {d}x -\frac{a}{2}\int _{{\mathbb {R}}^3}\left( |x|^{-1}*u^2\right) u^2\hbox {d}x \end{aligned}$$ with the mass constraint $$\Vert u\Vert _{L^2({\mathbb {R}}^3)}=1$$ , for $$m\in {\mathbb {R}}$$ , $$a>0$$ and V(x) is a nonpositive singular potential. Our main results are the existence and blow-up behavior of minimizers when a tends to a critical value $$a^*=\Vert Q\Vert ^2_{L^2({\mathbb {R}}^3)}$$ , where Q is a positive ground state of the following problem $$\begin{aligned} \sqrt{-\Delta }u+u-\left( |x|^{-1}*u^2\right) u=0\quad \text{ in }\, {\mathbb {R}}^3,\quad u\in H^{\frac{1}{2}}({\mathbb {R}}^3). \end{aligned}$$

Vortex-type solutions for magnetic pseudo-relativistic Hartree equation

We deal with a class of magnetic pseudo-relativistic Hartree type equation where , m>0, is a continuous vector potential, is an external continuous scalar potential and is a convolution kernel, is a positive constant, , . Under the action of some subgroup of linear isometries on potential A and W, and some assumptions on the decay of A and W at infinity, we prove the existence of vortex-type solutions to this problem by using variational methods and asymptotic estimates as p = 2.

Nondegeneracy of solutions for a critical Hartree equation

The aim of this paper is to prove the nondegeneracy of the unique positive solutions for the following critical Hartree type equations when μ>0 is close to 0, −Δu=Iμ∗u2μ∗u2μ∗−1,x∈RN, where Iμ(x)=Γ(μ2)Γ(N−μ2)πN22N−μ|x|μ is the Riesz potential and 2μ∗=2N−μN−2 is the upper critical exponent due to the Hardy–Littlewood–Sobolev inequality.

Stability of equilibria for a Hartree equation for random fields

We consider a Hartree equation for a random field, which describes the temporal evolution of infinitely many fermions. On the Euclidean space, this equation possesses equilibria which are not localized. We show their stability through a scattering result, with respect to localized perturbations in the not too focusing case in high dimensions d≥4. This provides an analogue of the results of Lewin and Sabin [22], and of Chen, Hong and Pavlović [11] for the Hartree equation on operators. The proof relies on dispersive techniques used for the study of scattering for the nonlinear Schrödinger and Gross-Pitaevskii equations.

On critical pseudo-relativistic Hartree equation with potential well

The aim of this paper is to investigate the existence and asymptotic behavior of the solutions for the critical pseudo-relativistic Hartree equation $$ \sqrt{-\Delta+m^{2}}u+(\beta V(x)-\lambda)u =\bigg(\int_{\mathbb{R}^{N}}\frac{|u(z)|^{2_{\mu}^{\ast}}}{|x-z|^{\mu}}dz\bigg)|u| ^{2_{\mu}^{\ast}-2}u $$% for $\mathbb{R}^{N}$, where $m, \lambda, \beta\in\mathbb{R}^+$, $0&lt; \mu&lt; N$, $N\geq3$, $2_{\mu}^{\ast}=({2N-\mu})/({N-1})$ plays the role of critical exponent due to the Hardy-Littlewood-Sobolev inequality. By transforming the nonlocal problem into a local one via the Dirichlet-to-Neumann map, we are able to obtain the existence of the solutions by variational methods. Suppose that $0&lt; \lambda&lt; \lambda_{1}(\Omega)$ with $\lambda_{1}(\Omega)$ the first eigenvalue and the parameter $\beta$ is large enough, we can prove the existence of ground state solutions. Furthermore, for any sequences $\beta_{n}\rightarrow\infty$, we can show that the ground state solutions $\{u_{n}\}$ converges to a solution of $$ \sqrt{-\Delta+m^{2}}u-\lambda u= \bigg(\int_{\Omega}\frac{|u(z)|^{2_{\mu}^{\ast}}}{|x-z|^{\mu}}dz\bigg) |u|^{2_{\mu}^{\ast}-2}u \quad \mbox{in } \Omega, $$ where $\Omega :=\mbox{int} V^{-1}(0)$ is a nonempty bounded set with smooth boundary. By the way we also establish the existence and nonexistence results for the ground state solutions of the problems set on bounded domain.

Below and beyond the mass–energy threshold: scattering for the Hartree equation with radial data in $$\pmb {d \ge 5}$$

We consider the Cauchy problem of the focusing $$\dot{H}^{1/2}$$-critical Hartree equation $$\begin{aligned} i u_{t} + \Delta u = - \Bigl ( |\cdot |^{-3} *|u(t)|^{2} \Bigr )u(t,x), \quad (t,x)\in {\mathbb {R}} \times {\mathbb {R}}^{d}. \end{aligned}$$By adapting the methods in Dodson and Murphy (Proc Am Math Soc 145(11):4859–4867, 2017), we shall prove a scattering result for solutions both below and beyond the mass–energy threshold M(Q)E(Q) and uniformly describe both cases the boundary of the scattering region by the ground state’s mass and potential energy product.