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Smoothing properties and retarded estimates for some dispersive evolution equations

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Abstract

Smoothing properties, in the form of space-time integrability properties, play an important role in the study of dispersive evolution equations. A number of them follow from a combination of general arguments and specific estimates. We present a general formulation which makes the separation between the two types of ingredients as clear as possible, and we illustrate it with the examples of the Schrödinger equation, of the wave equation, and of a class of 1+1 dimensional equations related to the Benjamin-Ono equation. Of special interest for the Cauchy problem are retarded estimates expressed in terms of those properties. We derive a number of such estimates associated with the last example, and we mention briefly an application of those estimates to the Cauchy problem for the generalized Benjamin-Ono equation.

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Authors and Affiliations

  1. Laboratoire de Physique Théorique et Hautes Energies, Université de Paris XI, Bâtiment 211, F-91405, Orsay, France

    J. Ginibre

  2. Dipartimento di Fisica, Università di Bologna and INFN, Sezione di Bologna, Italy

    G. Velo

Authors
  1. J. Ginibre
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  2. G. Velo
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Communicate by H. Araki

Laboratoire associé au Centre National de la Recherche Scientifique

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Ginibre, J., Velo, G. Smoothing properties and retarded estimates for some dispersive evolution equations. Commun.Math. Phys. 144, 163–188 (1992). https://doi.org/10.1007/BF02099195

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  • DOI: https://doi.org/10.1007/BF02099195

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