Quasi-periodic solutions with Sobolev regularity of NLS on $\mathbb T^d$ with a multiplicative potential

Massimiliano Berti, Philippe Bolle
2013 Journal of the European Mathematical Society (Print)  
We prove the existence of quasi-periodic solutions for Schrödinger equations with a multiplicative potential on T d , d ≥ 1, finitely differentiable nonlinearities, and tangential frequencies constrained along a pre-assigned direction. The solutions have only Sobolev regularity both in time and space. If the nonlinearity and the potential are C ∞ then the solutions are C ∞ . The proofs are based on an improved Nash-Moser iterative scheme, which assumes the weakest tame estimates for the inverse
more » ... linearized operators ("Green functions") along scales of Sobolev spaces. The key off-diagonal decay estimates of the Green functions are proved via a new multiscale inductive analysis. The main novelty concerns the measure and "complexity" estimates. Mathematics Subject Classification (2010): 35Q55, 37K55, 37K50 Quasi-periodic solutions of NLS on T d 231 Problem 2 has often been bypassed by considering pseudo-differential PDEs, substituting the multiplicative potential V (x) by a "convolution potential" V * (e ij ·x ) = m j e ij ·x , m j ∈ R, j ∈ Z d , which, by definition, is diagonal on the exponentials. The scalars m j are called Fourier multipliers. Concerning problem 1, since the approach of Craig-Wayne and Bourgain requires only the first order Melnikov nonresonance conditions, it works well, in principle, in the case of multiple eigenvalues, in particular for PDEs in higher spatial dimensions. Actually the first existence results for periodic solutions of NLW and NLS on T d , d ≥ 2, have been established by Bourgain in [7]- [10] . Here the singular sites form huge clusters (not only points as in d = 1) but are still "separated at infinity". The nonlinearities are polynomial and the solutions have Gevrey regularity in space and time. Recently these results were extended in [2]-[5] to prove the existence of periodic solutions, with only Sobolev regularity, for NLS and NLW in any dimension and with finitely differentiable nonlinearities. Actually in [4], [5] the PDEs are defined not only on tori, but on any compact Zoll manifold, Lie group or homogeneous space. These results are proved via an abstract Nash-Moser implicit function theorem (a simple Newton method is not sufficient). Clearly, a difficulty when working with functions having only Sobolev regularity is that the Green functions will exhibit only a polynomial decay off the diagonal, and not exponential (or subexponential). A key concept one must exploit is the interpolation/tame estimates. For PDEs on Lie groups only weak properties of "localization" (ii) of the eigenfunctions hold (see [5] ). Nevertheless these properties imply block diagonal decay for the matrix which represents the multiplication operator in the eigenfunctions basis, sufficient to achieve the tame estimates. We also mention that existence of periodic solutions for NLS on T d has been proved, for analytic nonlinearities, by Gentile-Procesi [26] via Lindstedt series techniques, and, in the differentiable case, by Delort [18] using paradifferential calculus. Regarding quasi-periodic solutions, Bourgain [10] was the first to prove their existence for PDEs in higher dimensions, actually for nonlinear Schrödinger equations with Fourier multipliers and polynomial nonlinearities on T d with d = 2. The Fourier multipliers, in number equal to the tangential frequencies of the quasi-periodic solution, play the role of external parameters. The main difficulty arises in the multiscale argument to estimate the decay of the Green functions. Due to the degeneracy of the eigenvalues of the Laplacian, the singular submatrices that one has to control are huge. If d = 2, careful estimates on the number of integer vectors on a sphere allowed anyway Bourgain to show that the required nonresonance conditions are fulfilled for "most" Fourier multipliers. More recently Bourgain [13] improved the techniques of [10] by proving the existence of quasi-periodic solutions for nonlinear wave and Schrödinger equations with Fourier multipliers on any T d , d > 2, still for polynomial nonlinearities. The improvement in [13] comes from the use of sophisticated techniques developed in the context of Anderson localization theory in Bourgain-Goldstein-Schlag [14] and Bourgain [11] (see also ). These techniques (subharmonic functions, Cartan theorem,
doi:10.4171/jems/361 fatcat:dz2sgxq7efdchcasliktpjnfzm