scholarly journals Global well-posedness and scattering for the defocusingH12-subcritical Hartree equation inRd

2009 ◽  
Vol 26 (5) ◽  
pp. 1831-1852 ◽  
Author(s):  
Changxing Miao ◽  
Guixiang Xu ◽  
Lifeng Zhao
Keyword(s):  
Hartree Equation ◽  
Well Posedness

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

2020 ◽  
Vol 17 (04) ◽  
pp. 727-763
Author(s):  
Anudeep Kumar Arora ◽  
Svetlana Roudenko
Keyword(s):  
Finite Time ◽  
Blow Up ◽  
Hartree Equation ◽  
Energy Threshold ◽  
Local Theory ◽  
Positive Energy ◽  
Small Data ◽  
Well Posedness ◽  
Time Behavior ◽  
Mass Energy

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.

Global well-posedness and blow-up for the hartree equation

2017 ◽  
Vol 37 (4) ◽  
pp. 941-948 ◽  
Author(s):  
Lingyan YANG ◽  
Xiaoguang LI ◽  
Yonghong WU ◽  
Louis CACCETTA
Keyword(s):  
Blow Up ◽  
Hartree Equation ◽  
Well Posedness

scholarly journals Well-posedness in weighted spaces for the generalized Hartree equation with p < 2

2021 ◽  
pp. 2150074
Author(s):  
Anudeep K. Arora ◽  
Oscar Riaño ◽  
Svetlana Roudenko
Keyword(s):  
Global Existence ◽  
Local Existence ◽  
Weighted Sobolev Spaces ◽  
Riesz Transform ◽  
Hartree Equation ◽  
Positive Energy ◽  
Well Posedness ◽  
Lebesgue Spaces ◽  
Positive Time ◽  
Well Posed

We investigate the well-posedness in the generalized Hartree equation [Formula: see text], [Formula: see text], [Formula: see text], for low powers of nonlinearity, [Formula: see text]. We establish the local well-posedness for a class of data in weighted Sobolev spaces, following ideas of Cazenave and Naumkin, Local existence, global existence, and scattering for the nonlinear Schrödinger equation, Comm. Contemp. Math. 19(2) (2017) 1650038. 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 [Formula: see text]-supercritical setting we obtain a subset of locally well-posed data with positive energy, which blows up in finite time.

scholarly journals Global well-posedness for the defocusing Hartree equation with radial data in ℝ4

2019 ◽  
Vol 22 (02) ◽  
pp. 1950004 ◽  
Author(s):  
Changxing Miao ◽  
Guixiang Xu ◽  
Jianwei-Urbain Yang
Keyword(s):  
Open Problem ◽  
Critical Case ◽  
Strichartz Estimate ◽  
Hartree Equation ◽  
Well Posedness ◽  
Schrodinger Operator ◽  
Smoothing Effect ◽  
Local Smoothing ◽  
Global Result ◽  
Long Time

By [Formula: see text]-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 [Formula: see text] where [Formula: see text], and [Formula: see text], [Formula: see text], with radial data in [Formula: see text] for [Formula: see text]. It is a sharp global result except the critical case [Formula: see text], which is a very difficult open problem.

scholarly journals On the focusing generalized Hartree equation

2020 ◽  
Vol 9999 (9999) ◽  
pp. 1-20
Author(s):  
Anudeep Kumar Arora ◽  
Svetlana Roudenko ◽  
Kai Yang
Keyword(s):  
Global Existence ◽  
Finite Time ◽  
Recent Progress ◽  
Blow Up ◽  
Riesz Potential ◽  
Type Equation ◽  
Hartree Equation ◽  
Well Posedness ◽  
Nonlocal Nonlinearity ◽  
Numerical Investigations

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.

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