Nash–Moser iteration and singular perturbations
Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 4, pp. 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 $ϵ\to 0$. 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{\textendash}Moser iteration and singular perturbations},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {499--527},
publisher = {Elsevier},
volume = {28},
number = {4},
year = {2011},
doi = {10.1016/j.anihpc.2011.05.001},
zbl = {1237.47066},
mrnumber = {2823882},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2011.05.001/}
}
TY  - JOUR
AU  - Texier, Benjamin
AU  - Zumbrun, Kevin
TI  - Nash–Moser iteration and singular perturbations
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2011
DA  - 2011///
SP  - 499
EP  - 527
VL  - 28
IS  - 4
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2011.05.001/
UR  - https://zbmath.org/?q=an%3A1237.47066
UR  - https://www.ams.org/mathscinet-getitem?mr=2823882
UR  - https://doi.org/10.1016/j.anihpc.2011.05.001
DO  - 10.1016/j.anihpc.2011.05.001
LA  - en
ID  - AIHPC_2011__28_4_499_0
ER  - 
%0 Journal Article
%A Texier, Benjamin
%A Zumbrun, Kevin
%T Nash–Moser iteration and singular perturbations
%J Annales de l'I.H.P. Analyse non linéaire
%D 2011
%P 499-527
%V 28
%N 4
%I Elsevier
%U https://doi.org/10.1016/j.anihpc.2011.05.001
%R 10.1016/j.anihpc.2011.05.001
%G en
%F 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://archive.numdam.org/articles/10.1016/j.anihpc.2011.05.001/

[1] S. Alinhac, P. Gérard, Opérateurs pseudo-différentiels et théorème de Nash–Moser, Savoirs Actuels, InterEditions/éditions du CNRS, Paris/Meudon (1991) | MR

[2] B. Alvarez-Samaniego, D. Lannes, A Nash–Moser theorem for singular evolution equations. Application to the Serre and Green–Naghdi equations, Indiana Univ. Math. J. 57 no. 1 (2008), 97-131 | MR | Zbl

[3] A. Dressel, W.-A. Yong, Existence of traveling-wave solutions for hyperbolic systems of balance laws, Arch. Ration. Mech. Anal. 182 no. 1 (2006), 49-75 | MR | Zbl

[4] R.S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.) 7 no. 1 (1982), 65-222 | MR | Zbl

[5] G. Iooss, P. Plotnikov, J. Toland, Standing waves on an infinitely deep perfect fluid under gravity, Arch. Ration. Mech. Anal. 177 no. 3 (2005), 367-478 | MR | Zbl

[6] J.-L. Joly, G. Métivier, J. Rauch, Transparent nonlinear geometric optics and Maxwell–Bloch equations, J. Differential Equations 166 (2000), 175-250 | MR | Zbl

[7] D. Lannes, Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators, J. Funct. Anal. 232 no. 2 (2006), 495-539 | MR | Zbl

[8] J. Nash, The imbedding problem for Riemannian manifolds, Ann. of Math. (2) 63 (1956), 20-63 | MR | Zbl

[9] C. Mascia, K. Zumbrun, Pointwise Greenʼs function bounds and stability of relaxation shocks, Indiana Univ. Math. J. 51 no. 4 (2002), 773-904 | MR | Zbl

[10] C. Mascia, K. Zumbrun, Stability of large-amplitude shock profiles of general relaxation systems, SIAM J. Math. Anal. 37 no. 3 (2005), 889-913 | MR | Zbl

[11] C. Mascia, K. Zumbrun, Spectral stability of weak relaxation shock profiles, Comm. Partial Differential Equations 34 no. 1–3 (2009), 119-136 | MR | Zbl

[12] G. Métivier, Para-Differential Calculus and Applications to the Cauchy Problem for Nonlinear Systems, Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series vol. 5, Edizioni della Normale, Pisa (2008) | MR | Zbl

[13] G. Métivier, J. Rauch, Dispersive stabilization, Bull. Lond. Math. Soc. 42 no. 2 (2010), 250-262 | MR | Zbl

[14] G. Métivier, K. Zumbrun, Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems, Mem. Amer. Math. Soc. 175 no. 826 (2005) | MR | Zbl

[15] G. Métivier, K. Zumbrun, Existence of semilinear relaxation shocks, J. Math. Pures Appl. 92 no. 3 (September 2009), 209-231 | MR

[16] G. Métivier, K. Zumbrun, Existence and sharp localization in velocity of small-amplitude Boltzmann shocks, Kinet. Relat. Models 2 no. 4 (December 2009), 667-705 | MR

[17] G. Métivier, B. Texier, K. Zumbrun, Existence of quasilinear relaxation shock profiles, Ann. Sci. Fac. Sci. Toulouse (2011), in press. | Numdam | MR

[18] J. Moser, A rapidly convergent iteration method and non-linear partial differential equations. I, Ann. Scuola Norm. Sup. Pisa (3) 20 (1966), 265-315 J. Moser, A rapidly convergent iteration method and non-linear partial differential equations. II, Ann. Scuola Norm. Sup. Pisa (3) 20 (1966), 499-535 | EuDML | Numdam | MR | Zbl

[19] B.P. Rynne, M.A. Youngson, Linear Functional Analysis, Springer Undergraduate Mathematics Series, Springer-Verlag London, Ltd., London (2008) | MR

[20] B. Texier, Derivation of the Zakharov equations, Arch. Ration. Mech. Anal. 184 no. 1 (2007), 121-183 | MR | Zbl

[21] B. Texier, K. Zumbrun, Galloping instability of viscous shock waves, Physica D 237 no. 10–12 (2008), 1553-1601 | MR | Zbl

[22] X. Saint-Raymond, A simple Nash–Moser implicit function theorem, Enseign. Math. (2) 35 no. 3–4 (1989), 217-226 | MR | Zbl

[23] W.-A. Yong, K. Zumbrun, Existence of relaxation shock profiles for hyperbolic conservation laws, SIAM J. Appl. Math. 60 no. 5 (2000), 1565-1575 | MR | Zbl

Cited by Sources: