# American Institute of Mathematical Sciences

July  2015, 14(4): 1357-1376. doi: 10.3934/cpaa.2015.14.1357

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

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

Received  March 2013 Revised  November 2013 Published  April 2015

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.
Citation: Jean Ginibre, Giorgio Velo. Modified wave operators without loss of regularity for some long range Hartree equations. II. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1357-1376. doi: 10.3934/cpaa.2015.14.1357
##### References:
 [1] J. Bergh and J. Löfström, Interpolation Spaces,, Springer, (1976). Google Scholar [2] J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree-type equations I,, \emph{Rev. Math. Phys.}, 12 (2000), 361. doi: 10.1142/S0129055X00000137. Google Scholar [3] J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree-type equations II,, \emph{Ann. Henri Poincar\'e}, 1 (2000), 753. doi: 10.1007/PL00001014. Google Scholar [4] J. Ginibre and G. Velo, Long range scattering for the Wave-Schrödinger system revisited,, \emph{J. Diff. Eq.}, 252 (2012), 1642. doi: 10.1016/j.jde.2011.07.003. Google Scholar [5] J. Ginibre and G. Velo, Modified wave operators without loss of regularity for some long range Hartree equations I,, preprint, (2012). doi: 10.1007/s00023-013-0257-5. Google Scholar [6] K. Nakanishi, Modified wave operators for the Hartree equation with data, image and convergence in the same space,, \emph{Commun. Pure Appl. Anal.}, 1 (2002), 237. doi: 10.3934/cpaa.2002.1.237. Google Scholar [7] K. Nakanishi, Modified wave operators for the Hartree equation with data, image and convergence in the same space II,, \emph{Ann. Henri Poincar\'e}, 3 (2002), 503. doi: 10.1007/s00023-002-8626-5. Google Scholar [8] E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Univ. Press, (1970). Google Scholar

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##### References:
 [1] J. Bergh and J. Löfström, Interpolation Spaces,, Springer, (1976). Google Scholar [2] J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree-type equations I,, \emph{Rev. Math. Phys.}, 12 (2000), 361. doi: 10.1142/S0129055X00000137. Google Scholar [3] J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree-type equations II,, \emph{Ann. Henri Poincar\'e}, 1 (2000), 753. doi: 10.1007/PL00001014. Google Scholar [4] J. Ginibre and G. Velo, Long range scattering for the Wave-Schrödinger system revisited,, \emph{J. Diff. Eq.}, 252 (2012), 1642. doi: 10.1016/j.jde.2011.07.003. Google Scholar [5] J. Ginibre and G. Velo, Modified wave operators without loss of regularity for some long range Hartree equations I,, preprint, (2012). doi: 10.1007/s00023-013-0257-5. Google Scholar [6] K. Nakanishi, Modified wave operators for the Hartree equation with data, image and convergence in the same space,, \emph{Commun. Pure Appl. Anal.}, 1 (2002), 237. doi: 10.3934/cpaa.2002.1.237. Google Scholar [7] K. Nakanishi, Modified wave operators for the Hartree equation with data, image and convergence in the same space II,, \emph{Ann. Henri Poincar\'e}, 3 (2002), 503. doi: 10.1007/s00023-002-8626-5. Google Scholar [8] E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Univ. Press, (1970). Google Scholar
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