Modified wave operators without loss of regularity for some long range Hartree equations. II

Jean Ginibre; Giorgio Velo

doi:10.3934/cpaa.2015.14.1357

Article Contents

2015, Volume 14, Issue 4: 1357-1376. Doi: 10.3934/cpaa.2015.14.1357

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Modified wave operators without loss of regularity for some long range Hartree equations. II

Jean Ginibre^1, and
Giorgio Velo^2,

1.
Laboratoire de Physique Théorique, (Unité Mixte de Recherche CNRS UMR 8627), Université de Paris-Sud, Bâtiment 210, F-91405 Orsay Cedex
2.
Dipartimento di Fisica e Astronomia, Università di Bologna and INFN, Sezione di Bologna

Received: March 2013

Revised: November 2013

Published: July 2015

Abstract / Introduction Related Papers Cited by

Abstract

We continue the study of the theory of scattering for some long range Hartree equations with potential $|x|^{-\gamma}$, performed in a previous paper, denoted as I, in the range $1/2 < \gamma < 1$. Here we extend the results to the range $1/3 < \gamma < 1/2$. More precisely we study the local Cauchy problem with infinite initial time, which is the main step in the construction of the modified wave operators. We solve that problem without loss of regularity between the asymptotic state and the solution, as in I, but in contrast to I, we are no longer able to cover the entire subcriticality range of regularity of the solutions. The method is an extension of that of I, using a better approximate asymptotic form of the solutions obtained as the next step of a natural procedure of successive approximations.

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Mathematics Subject Classification: Primary: 35P25; Secondary: 35B40, 35Q40, 81U99.

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References

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