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Scattering Below Critical Energy for the Radial 4D Yang-Mills Equation and for the 2D Corotational Wave Map System

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Abstract

Given g and f  =  gg′, we consider solutions to the following non linear wave equation :

$$\left\{ \begin{array}{l}\displaystyle u_{tt} - u_{rr} - \frac 1 r u_r = - \frac{f(u)}{r^2},\\(u, u_t)|_{t =0} = (u_0,u_1). \end{array}\right.$$

Under suitable assumptions on g, this equation admits non-constant stationary solutions : we denote Q one with least energy. We characterize completely the behavior as time goes to  ±∞ of solutions (u, u t ) corresponding to data with energy less than or equal to the energy of Q : either it is (Q, 0) up to scaling, or it scatters in the energy space.

Our results include the cases of the 2 dimensional corotational wave map system, with target \({{\mathbb S}^2}\) , in the critical energy space, as well as the 4 dimensional, radially symmetric Yang-Mills fields on Minkowski space, in the critical energy space.

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Authors and Affiliations

  1. Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128, Palaiseau Cedex, France

    Raphaël Côte

  2. Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, IL, 60637-1514, USA

    Carlos E. Kenig

  3. Département de Mathématiques, Université de Cergy-Pontoise / Saint-Martin, 2, avenue Adolphe Chauvin, 95 302, Cergy-Pontoise Cedex, France

    Frank Merle

Authors
  1. Raphaël Côte
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  2. Carlos E. Kenig
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  3. Frank Merle
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Correspondence to Raphaël Côte.

Additional information

Communicated by P. Constantin

Centre National de la Recherche Scientifique.

Institut des Hautes Études Scientifiques.

The work of R.C. and F.M. has been supported in part by ANR grant ONDE NONLIN, and the work of C.E.K. has been supported in part by NSF.

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Côte, R., Kenig, C.E. & Merle, F. Scattering Below Critical Energy for the Radial 4D Yang-Mills Equation and for the 2D Corotational Wave Map System. Commun. Math. Phys. 284, 203–225 (2008). https://doi.org/10.1007/s00220-008-0604-4

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  • DOI: https://doi.org/10.1007/s00220-008-0604-4

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