Quasi-periodic solutions of PDEs

Berti, Massimiliano

Séminaire Laurent Schwartz — EDP et applications (2011-2012), Talk no. 30, 11 p.

The aim of this talk is to present some recent existence results about quasi-periodic solutions for PDEs like nonlinear wave and Schrödinger equations in ${𝕋}^d$, $d\ge 2$, and the $1$-$d$ derivative wave equation. The proofs are based on both Nash-Moser implicit function theorems and KAM theory.

[1] Bambusi D., Berti M., Magistrelli E., Degenerate KAM theory for partial differential equations, J. Differential Equations 250, 3379-3397, 2011. | MR | Zbl

[2] Bambusi D., Delort J.M., Grebért B., Szeftel J., Almost global existence for Hamiltonian semilinear Klein-Gordon equations with small Cauchy data on Zoll manifolds, Comm. Pure Appl. Math. 60, 11, 1665-1690, 2007. | MR | Zbl

[3] Berti M., Nonlinear Oscillations of Hamiltonian PDEs, Progr. Nonlinear Differential Equations Appl. 74, H. Brézis, ed., Birkhäuser, Boston, 1-181, 2008. | MR | Zbl

[4] Berti M., Biasco L., Branching of Cantor manifolds of elliptic tori and applications to PDEs, Comm. Math. Phys, 305, 3, 741-796, 2011. | MR | Zbl

[5] Berti M., Biasco L., Procesi M. KAM theory for the Hamiltonian derivative wave equation, preprint 2011.

[6] Berti M., Bolle P., Quasi-periodic solutions with Sobolev regularity of NLS on ${𝕋}^d$ with a multiplicative potential, to appear on the Journal European Math. Society. | MR

[7] Berti M., Bolle P., Quasi-periodic solutions of nonlinear Schrödinger equations on ${𝕋}^d$, Rend. Lincei Mat. Appl. 22, 223-236, 2011. | MR | Zbl

[8] Berti M., Bolle P., Sobolev quasi periodic solutions of multidimensional wave equations with a multiplicative potential, preprint 2012. | MR

[9] Berti M., Bolle P., Procesi M., An abstract Nash-Moser theorem with parameters and applications to PDEs, Ann. I. H. Poincaré, 27, 377-399, 2010. | Numdam | MR | Zbl

[10] Berti M., Procesi M., Nonlinear wave and Schrödinger equations on compact Lie groups and homogeneous spaces, Duke Math. J., 159, 3, 479-538, 2011. | MR

[11] Bourgain J., Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE, Internat. Math. Res. Notices, no. 11, 1994. | MR | Zbl

[12] Bourgain J., On Melnikov’s persistency problem, Internat. Math. Res. Letters, 4, 445 - 458, 1997. | MR | Zbl

[13] Bourgain J., Quasi-periodic solutions of Hamiltonian perturbations of $2D$ linear Schrödinger equations, Annals of Math. 148, 363-439, 1998. | MR | Zbl

[14] Bourgain J., Periodic solutions of nonlinear wave equations, Harmonic analysis and partial differential equations, 69–97, Chicago Lectures in Math., Univ. Chicago Press, 1999. | MR | Zbl

[15] Bourgain J., Green’s function estimates for lattice Schrödinger operators and applications, Annals of Mathematics Studies 158, Princeton University Press, Princeton, 2005. | MR | Zbl

[16] Colliander J., Keel M., Staffilani G., Takaoka H., Tao T., Weakly turbolent solutions for the cubic defocusing nonlinear Schrödinger equation, 181, 1, 39-113, Inventiones Math., 2010. | MR | Zbl

[17] Craig W., Problèmes de petits diviseurs dans les équations aux dérivées partielles, Panoramas et Synthèses, 9, Société Mathématique de France, Paris, 2000. | MR | Zbl

[18] Craig W., Wayne C. E., Newton’s method and periodic solutions of nonlinear wave equation, Comm. Pure Appl. Math. 46, 1409-1498, 1993. | MR | Zbl

[19] Eliasson L.H., Perturbations of stable invariant tori for Hamiltonian systems, Ann. Sc. Norm. Sup. Pisa., 15, 115-147, 1988. | Numdam | MR | Zbl

[20] Eliasson L. H., Kuksin S., KAM for nonlinear Schrödinger equation, Annals of Math., 172, 371-435, 2010. | MR | Zbl

[21] Geng J., You J., A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces, Comm. Math. Phys. 262, 343-372, 2006. | MR | Zbl

[22] Grebert B., Thomann L., KAM for the quantum harmonic oscillator, Comm. Math. Phys. 307, 2, 383-427, 2011. | MR | Zbl

[23] Kappeler T., Pöschel J., KAM and KdV, Springer, 2003.

[24] Kuksin S., Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum, Funktsional Anal. i Prilozhen. 2, 22-37, 95, 1987. | MR | Zbl

[25] Kuksin S., Analysis of Hamiltonian PDEs, Oxford Lecture series in Math. and its applications, 19, Oxford University Press, 2000. | MR | Zbl

[26] Liu J., Yuan X., A KAM Theorem for Hamiltonian Partial Differential Equations with Unbounded Perturbations, Comm. Math. Phys, 307 (3), 629-673, 2011. | MR | Zbl

[27] Lojasiewicz S., Zehnder E., An inverse function theorem in Fréchet-spaces, J. Funct. Anal. 33, 165-174, 1979. | MR | Zbl

[28] Pöschel J., A KAM-Theorem for some nonlinear PDEs, Ann. Scuola Norm. Sup. Pisa, Cl. Sci., 23, 119-148, 1996. | Numdam | MR | Zbl

[29] Procesi C., Procesi M., A normal form for the Schrödinger equation with analytic non-linearities, to appear on Comm. Math.Phys. | MR

[30] Wang W. M., Supercritical nonlinear Schrödinger equations I: quasi-periodic solutions, preprint 2010.

[31] Wayne E., Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys. 127, 479-528, 1990. | MR | Zbl