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. 499–527DOI 10.1016/J.ANIHPC.2011.05.001