Well posedness and smoothing effect of Schrödinger-Poisson equation

In this work we take under consideration the Cauchy problem for the Schrödinger-Poisson type equation i t u=- x2 u+V (u) u-f (∫u∫2) u, where f represents a local nonlinear interaction (we take into account both attractive and repulsive models) and V is taken as a suitable solution of the Poisson equ...

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Autores principales: De Leo, Mariano Fernando, Rial, Diego Fernando
Publicado: 2007
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00222488_v48_n9_p_DeLeo
http://hdl.handle.net/20.500.12110/paper_00222488_v48_n9_p_DeLeo
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paper:paper_00222488_v48_n9_p_DeLeo
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spelling paper:paper_00222488_v48_n9_p_DeLeo2023-06-08T14:48:14Z Well posedness and smoothing effect of Schrödinger-Poisson equation De Leo, Mariano Fernando Rial, Diego Fernando In this work we take under consideration the Cauchy problem for the Schrödinger-Poisson type equation i t u=- x2 u+V (u) u-f (∫u∫2) u, where f represents a local nonlinear interaction (we take into account both attractive and repulsive models) and V is taken as a suitable solution of the Poisson equation V=12 ∫x∫ (C- ∫u∫2), C Cc∞ is the doping profile or impurities. We show that this problem is locally well posed in the weighted Sobolev spaces Hs { Hs (R): (1+ x2) 12 ∫∫2 <∞} with s1, which means the local existence, uniqueness, and continuity of the solution with respect to the initial data. Moreover, under suitable assumptions on the local interaction, we show the existence of global solutions. Finally, we establish that for s1 local in time and space, smoothing effects are present in the solution; more precisely, in this problem there is locally a gain of half a derivative. © 2007 American Institute of Physics. Fil:De Leo, M. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Rial, D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2007 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00222488_v48_n9_p_DeLeo http://hdl.handle.net/20.500.12110/paper_00222488_v48_n9_p_DeLeo
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
description In this work we take under consideration the Cauchy problem for the Schrödinger-Poisson type equation i t u=- x2 u+V (u) u-f (∫u∫2) u, where f represents a local nonlinear interaction (we take into account both attractive and repulsive models) and V is taken as a suitable solution of the Poisson equation V=12 ∫x∫ (C- ∫u∫2), C Cc∞ is the doping profile or impurities. We show that this problem is locally well posed in the weighted Sobolev spaces Hs { Hs (R): (1+ x2) 12 ∫∫2 <∞} with s1, which means the local existence, uniqueness, and continuity of the solution with respect to the initial data. Moreover, under suitable assumptions on the local interaction, we show the existence of global solutions. Finally, we establish that for s1 local in time and space, smoothing effects are present in the solution; more precisely, in this problem there is locally a gain of half a derivative. © 2007 American Institute of Physics.
author De Leo, Mariano Fernando
Rial, Diego Fernando
spellingShingle De Leo, Mariano Fernando
Rial, Diego Fernando
Well posedness and smoothing effect of Schrödinger-Poisson equation
author_facet De Leo, Mariano Fernando
Rial, Diego Fernando
author_sort De Leo, Mariano Fernando
title Well posedness and smoothing effect of Schrödinger-Poisson equation
title_short Well posedness and smoothing effect of Schrödinger-Poisson equation
title_full Well posedness and smoothing effect of Schrödinger-Poisson equation
title_fullStr Well posedness and smoothing effect of Schrödinger-Poisson equation
title_full_unstemmed Well posedness and smoothing effect of Schrödinger-Poisson equation
title_sort well posedness and smoothing effect of schrödinger-poisson equation
publishDate 2007
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00222488_v48_n9_p_DeLeo
http://hdl.handle.net/20.500.12110/paper_00222488_v48_n9_p_DeLeo
work_keys_str_mv AT deleomarianofernando wellposednessandsmoothingeffectofschrodingerpoissonequation
AT rialdiegofernando wellposednessandsmoothingeffectofschrodingerpoissonequation
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