Work

On the lifespan of solutions and control of high Sobolev norms for the completely resonant NLS on tori

Year: 2024

Type: article

Abstract: We consider a completely resonant nonlinear Schrödinger equation on the d-dimensional torus, for any d≥1, with polynomial nonlinearity of any degree 2p+1, p≥1, which is gauge and translation invariant. We study the behaviour of high Sobolev Hs-norms of solutions, s≥s1+1>d/2+2, whose initial datum u0∈Hs satisfies an appropriate smallness condition on its low Hs1 and L2-norms respectively. We prove a polynomial upper bound on the possible growth of the Sobolev norm Hs over finite but long time scale that is exponential in the regularity parameter s1. As a byproduct we get stability of the low Hs1-norm over such time interval. A key ingredient in the proof is the introduction of a suitable “modified energy” that provides an a priori upper bound on the growth. This is obtained by combining para-differential techniques and suitable tame estimates.

Source: Journal of Functional Analysis

DOI: 10.1016/j.jfa.2024.110648

Authors: Roberto Feola, Jessica Elisa Massetti

Institutions: Roma Tre University, University of Rome Tor Vergata

Language: English

Topic: Advanced Mathematical Physics Problems

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $2,670

Funders: Istituto Nazionale di Alta Matematica "Francesco Severi", Ministero dell’Istruzione, dell’Università e della Ricerca, Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni, Ministero dell'Istruzione e del Merito