Local well-posedness for quasi-linear NLS with large Cauchy data on the circle
Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 1, pp. 119-164.

We prove local in time well-posedness for a large class of quasilinear Hamiltonian, or parity preserving, Schrödinger equations on the circle. After a paralinearization of the equation, we perform several paradifferential changes of coordinates in order to transform the system into a paradifferential one with symbols which, at the positive order, are constant and purely imaginary. This allows to obtain a priori energy estimates on the Sobolev norms of the solutions.

DOI : 10.1016/j.anihpc.2018.04.003
Mots-clés : NLS, Quasi-linear PDEs, Para-differential calculus, Local wellposedness, Dispersive equations, Energy method 
@article{AIHPC_2019__36_1_119_0,
     author = {Feola, R. and Iandoli, F.},
     title = {Local well-posedness for quasi-linear {NLS} with large {Cauchy} data on the circle},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {119--164},
     publisher = {Elsevier},
     volume = {36},
     number = {1},
     year = {2019},
     doi = {10.1016/j.anihpc.2018.04.003},
     mrnumber = {3906868},
     zbl = {1430.35207},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2018.04.003/}
}
TY  - JOUR
AU  - Feola, R.
AU  - Iandoli, F.
TI  - Local well-posedness for quasi-linear NLS with large Cauchy data on the circle
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2019
SP  - 119
EP  - 164
VL  - 36
IS  - 1
PB  - Elsevier
UR  - https://www.numdam.org/articles/10.1016/j.anihpc.2018.04.003/
DO  - 10.1016/j.anihpc.2018.04.003
LA  - en
ID  - AIHPC_2019__36_1_119_0
ER  - 
%0 Journal Article
%A Feola, R.
%A Iandoli, F.
%T Local well-posedness for quasi-linear NLS with large Cauchy data on the circle
%J Annales de l'I.H.P. Analyse non linéaire
%D 2019
%P 119-164
%V 36
%N 1
%I Elsevier
%U https://www.numdam.org/articles/10.1016/j.anihpc.2018.04.003/
%R 10.1016/j.anihpc.2018.04.003
%G en
%F AIHPC_2019__36_1_119_0
Feola, R.; Iandoli, F. Local well-posedness for quasi-linear NLS with large Cauchy data on the circle. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 1, pp. 119-164. doi : 10.1016/j.anihpc.2018.04.003. https://www.numdam.org/articles/10.1016/j.anihpc.2018.04.003/

[1] Alazard, T.; Baldi, P. Gravity capillary standing water waves, Arch. Ration. Mech. Anal., Volume 217 (2015) no. 3, pp. 741–830 | DOI | MR | Zbl

[2] Alazard, T.; Baldi, P.; Han-Kwan, D. Control of water waves, J. Eur. Math. Soc. (JEMS), Volume 20 (2018) no. 3, pp. 657–745 | DOI | MR | Zbl

[3] Alinhac, S. Paracomposition et opérateurs paradifférentiels, Commun. Partial Differ. Equ. (1986) | DOI | MR | Zbl

[4] Baldi, P. Periodic solutions of fully nonlinear autonomous equations of Benjamin–Ono type, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 30 (2013) | DOI | Numdam | MR | Zbl

[5] Baldi, P.; Berti, M.; Haus, E.; Montalto, R. Time quasi-periodic gravity water waves in finite depth, 2017 (preprint) | arXiv | MR | Zbl

[6] Baldi, P.; Berti, M.; Montalto, R. KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation, Math. Ann., Volume 359 (2014) | DOI | MR | Zbl

[7] Baldi, P.; Berti, M.; Montalto, R. KAM for autonomous quasilinear perturbations of KdV, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 33 (2016) | DOI | Numdam | MR | Zbl

[8] Baldi, P.; Haus, E.; Montalto, R. Controllability of quasi-linear Hamiltonian NLS equations, J. Differ. Equ. (2017) | DOI | MR | Zbl

[9] Bambusi, D.; Grébert, B. Birkhoff normal form for partial differential equations with tame modulus, Duke Math. J., Volume 3 (2006), pp. 507 | DOI | MR | Zbl

[10] Berti, M.; Delort, J.-M. Almost global existence of solutions for capillarity-gravity water waves equations with periodic spatial boundary conditions, 2017 (preprint) | arXiv

[11] Berti, M.; Montalto, R. Quasi-Periodic Standing Wave Solutions of Gravity-Capillary Water Waves, Memoirs of the American Math. Society, MEMO 891, 2016 (to appear) | arXiv | MR | Zbl

[12] Cazenave, T. Semilinear Schrödinger Equations, vol. 10, Courant Lecture Notes, 2003 | MR | Zbl

[13] Christ, M. Illposedness of a Schrödinger equation with derivative nonlinearity https://math.berkeley.edu/~mchrist/Papers/dnls.ps (preprint)

[14] Delort, J.M. A quasi-linear Birkhoff normal forms method. Application to the quasi-linear Klein–Gordon equation on S1 , Astérisque, Volume 341 (2012) | Numdam | MR | Zbl

[15] Delort, J.M. Quasi-Linear Perturbations of Hamiltonian Klein–Gordon Equations on Spheres, American Mathematical Society, 2015 | DOI | MR | Zbl

[16] Feola, R. KAM for a quasi-linear forced Hamiltonian NLS, 2015 (preprint) | arXiv

[17] Feola, R.; Procesi, M. Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations, J. Differ. Equ. (2014) | DOI | MR | Zbl

[18] Feola, R.; Procesi, M. KAM for quasi-linear autonomous NLS, 2017 (preprint) | arXiv

[19] Iooss, G.; Plotnikov, P.I.; Toland, J.F. Standing waves on an infinitely deep perfect fluid under gravity, Arch. Ration. Mech. Anal., Volume 177 (2005) no. 3, pp. 367–478 | DOI | MR | Zbl

[20] Kenig, C.E.; Ponce, G.; Vega, L. The Cauchy problem for quasi-linear Schrödinger equations, Invent. Math., Volume 158 (2004) no. 2, pp. 343–388 | DOI | MR | Zbl

[21] Métivier, G. Para-Differential Calculus and Applications to the Cauchy Problem for Nonlinear Systems, vol. 5, Edizioni della Normale, 2008 | MR | Zbl

[22] Moser, J. Rapidly convergent iteration method and non-linear partial differential equations – i, Ann. Sc. Norm. Super. Pisa, Volume 20 (1966) no. 2, pp. 265–315 | Numdam | MR | Zbl

[23] Poppenberg, M. Smooth solutions for a class of fully nonlinear Schrödinger type equations, Nonlinear Anal., Theory Methods Appl., Volume 45 (2001) no. 6, pp. 723–741 | DOI | MR | Zbl

[24] Taylor, M. Tools for PDE, Amer. Math. Soc., 2007 | DOI

[25] Zhang, J.; Gao, M.; Yuan, X. KAM tori for reversible partial differential equations, Nonlinearity, Volume 24 (2011) | DOI | MR | Zbl

Cité par Sources :