Nash–Moser iteration and singular perturbations

  • Benjamin Texier

    Université Paris Diderot (Paris 7), Institut de Mathématiques de Jussieu, UMR CNRS 7586, Paris, France
  • Kevin Zumbrun

    Indiana University, Bloomington, IN 47405, United States


We present a simple and easy-to-use Nash–Moser iteration theorem tailored for singular perturbation problems admitting a formal asymptotic expansion or other family of approximate solutions depending on a parameter . The novel feature is to allow loss of powers of ε as well as the usual loss of derivatives in the solution operator for the associated linearized problem. We indicate the utility of this theorem by describing sample applications to (i) systems of quasilinear Schrödinger equations, and (ii) existence of small-amplitude profiles of quasilinear relaxation systems in the degenerate case that the velocity of the profile is a characteristic mode of the hyperbolic operator.

Cite this article

Benjamin Texier, Kevin Zumbrun, Nash–Moser iteration and singular perturbations. Ann. Inst. H. Poincaré Anal. Non Linéaire 28 (2011), no. 4, pp. 499–527

DOI 10.1016/J.ANIHPC.2011.05.001