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

2020 ◽  
Vol 71 (2) ◽  
Author(s):  
Yanfang Gao ◽  
Zhiyong Wang
Keyword(s):  
Hartree Equation ◽  
Energy Threshold ◽  
Mass Energy

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.

Derivation of the photon mass-energy threshold

2005 ◽  
Author(s):  
Riccardo C. Storti ◽  
Todd J. Desiato
Keyword(s):  
Energy Threshold ◽  
Photon Mass ◽  
Mass Energy

scholarly journals Mass-energy threshold dynamics for dipolar quantum gases

2022 ◽  
Vol 20 (1) ◽  
pp. 165-200
Author(s):  
Van Duong Dinh ◽  
Luigi Forcella ◽  
Hichem Hajaiej
Keyword(s):  
Quantum Gases ◽  
Energy Threshold ◽  
Threshold Dynamics ◽  
Mass Energy

DETERMINISTIC APPROACH TO WATERSHED MODELING

1971 ◽  
Vol 2 (3) ◽  
pp. 146-166 ◽  
Author(s):  
DAVID A. WOOLHISER
Keyword(s):  
Boundary Conditions ◽  
Watershed Modeling ◽  
Deterministic Approach ◽  
Mass Energy ◽  
Deterministic Models ◽  
Conservation Of Mass ◽  
Hydrologic Process ◽  
Theoretical Structure ◽  
Physically Based ◽  
Initial And Boundary Conditions

Physically-based, deterministic models, are considered in this paper. Physically-based, in that the models have a theoretical structure based primarily on the laws of conservation of mass, energy, or momentum; deterministic in the sense that when initial and boundary conditions and inputs are specified, the output is known with certainty. This type of model attempts to describe the structure of a particular hydrologic process and is therefore helpful in predicting what will happen when some change occurs in the system.

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