Abstract
We reconsider the theory of scattering for some long-range Hartree equations with potential |x|−γ with 1/2 < γ < 1. 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 in the whole subcritical range without loss of regularity between the asymptotic state and the solution, thereby recovering a result of Nakanishi. Our method starts from a different parametrization of the solutions, already used in our previous papers. This reduces the proofs to energy estimates and avoids delicate phase estimates.
Article PDF
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Avoid common mistakes on your manuscript.
References
Bergh J., Löfström J.: Interpolation Spaces. Springer, Berlin (1976)
Ginibre J., Velo G.: Long range scattering and modified wave operators for some Hartree-type equations I. Rev. Math. Phys. 12, 361–429 (2000)
Ginibre J., Velo G.: Long range scattering and modified wave operators for some Hartree-type equations II. Ann. Henri Poincaré 1, 753–800 (2000)
Ginibre J., Velo G.: Long range scattering for the Wave-Schrödinger system revisited. J. Differ. Equ. 252, 1642–1667 (2012)
Nakanishi K.: Modified wave operators for the Hartree equation with data, image and convergence in the same space. Commun. Pure Appl. Anal. 1, 237–252 (2002)
Nakanishi K.: Modified wave operators for the Hartree equation with data, image and convergence in the same space II. Ann. Henri Poincaré 3, 503–535 (2002)
Stein E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Nader Masmoudi.
Rights and permissions
About this article
Cite this article
Ginibre, J., Velo, G. Modified Wave Operators Without Loss of Regularity for Some Long-Range Hartree Equations: I. Ann. Henri Poincaré 15, 829–862 (2014). https://doi.org/10.1007/s00023-013-0257-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-013-0257-5