Almost global existence for some nonlinear Schrödinger equations on \mathbbT^d in low regularity

2022/03/11 by Joackim Berniér, Bernier, Joackim, Benoît Grébert +1 · 1 citation
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2203.05799

openalex publication_date 2022/03/11 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28

Abstract

We are interested in the long time behavior of solutions of the nonlinear Schrödinger equation on the d-dimensional torus in low regularity, i.e. for small initial data in the Sobolev space H^s₀(\mathbb T^d) with s₀>d/2. We prove that, even in this context of low regularity, the H^s-norms, s≥ 0, remain under control during times, T_ε= exp (-(|logε|²)/(4log|logε|) ), exponential with respect to the initial size of the initial datum in H^s₀, ‖u(0)‖_H^s0=ε. For this, we add to the linear part of the equation a random Fourier multiplier in ℓ^∞(\mathbb Z^d) and show our stability result for almost any realization of this multiplier. In particular, with such Fourier multipliers, we obtain the almost global well posedness of the nonlinear Schrödinger equation on H^s₀(\mathbb T^d) for any s₀>d/2 and any d≥1.

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