Nash–Moser iteration and singular perturbations
Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 4, p. 499-527

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 ϵ0. 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.

@article{AIHPC_2011__28_4_499_0,
     author = {Texier, Benjamin and Zumbrun, Kevin},
     title = {Nash--Moser iteration and singular perturbations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {28},
     number = {4},
     year = {2011},
     pages = {499-527},
     doi = {10.1016/j.anihpc.2011.05.001},
     zbl = {1237.47066},
     mrnumber = {2823882},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2011__28_4_499_0}
}
Texier, Benjamin; Zumbrun, Kevin. Nash–Moser iteration and singular perturbations. Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 4, pp. 499-527. doi : 10.1016/j.anihpc.2011.05.001. http://www.numdam.org/item/AIHPC_2011__28_4_499_0/

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