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New bounds on the high Sobolev norms of the 1d NLS solutions

Diego Berti, Fabrice Planchon, Nikolay Tzvetkov, Nicola Visciglia

TL;DR

This work proves polynomial upper bounds on the growth of high Sobolev norms for solutions to the 1D periodic NLS on the torus. It introduces modified energies $\mathcal{E}_k$ built from structured densities, whose time derivatives lie in controlled linear spaces, and combines integration by parts with dispersive (and $X^{s,b}$) estimates to cancel growth terms. The main contributions include a simpler derivation of Bourgain's quintic bounds and an extension to higher nonlinearities, with sharp polynomial growth in $t$ for $k\neq 3m$ (epsilon-free) and a Bourgain-space-dependent treatment for $k=3m$. These results advance understanding of long-time dynamics for NLS on compact domains and provide quantitative control of high-regularity norms.

Abstract

We introduce modified energies that are suitable to get upper bounds on the high Sobolev norms for solutions to the $1$D periodic NLS. Our strategy is rather flexible and allows us to get a new and simpler proof of the bounds obtained by Bourgain in the case of the quintic nonlinearity, as well as its extension to the case of higher order nonlinearities. Our main ingredients are a combination of integration by parts and classical dispersive estimates.

New bounds on the high Sobolev norms of the 1d NLS solutions

TL;DR

This work proves polynomial upper bounds on the growth of high Sobolev norms for solutions to the 1D periodic NLS on the torus. It introduces modified energies built from structured densities, whose time derivatives lie in controlled linear spaces, and combines integration by parts with dispersive (and ) estimates to cancel growth terms. The main contributions include a simpler derivation of Bourgain's quintic bounds and an extension to higher nonlinearities, with sharp polynomial growth in for (epsilon-free) and a Bourgain-space-dependent treatment for . These results advance understanding of long-time dynamics for NLS on compact domains and provide quantitative control of high-regularity norms.

Abstract

We introduce modified energies that are suitable to get upper bounds on the high Sobolev norms for solutions to the D periodic NLS. Our strategy is rather flexible and allows us to get a new and simpler proof of the bounds obtained by Bourgain in the case of the quintic nonlinearity, as well as its extension to the case of higher order nonlinearities. Our main ingredients are a combination of integration by parts and classical dispersive estimates.
Paper Structure (8 sections, 11 theorems, 149 equations)

This paper contains 8 sections, 11 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Preliminary facts on the Cauchy theory
  3. The classical Sobolev setting
  4. The $X^{s,b}$ setting
  5. Construction of the energies ${\mathcal{E}}_k$
  6. Proof of Proposition \ref{['prop:structure']}
  7. Proof of Theorem \ref{['main']} for $k\neq 3m$
  8. Proof of Theorem \ref{['main']} for $k=3m$

Key Result

Theorem 1.1

Let $t_0\in{\mathbb R}$, $k,p\in \mathbb N$ with $k\geq 1$ and $p\geq 2$, $R>0$ and $\varepsilon>0$ be fixed. Then there exist $C,T>0$ and an energy with the structure such that the following property holds: if $u(t,x)\in {\mathcal{C}}({\mathbb R}; H^k({\mathbb T}))$ is the unique solution to NLSt0 with then we have:

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof : Sketch of Proof
  • Proposition 2.2
  • Definition 3.1
  • Remark 3.1
  • Definition 3.2
  • Definition 3.3
  • Remark 3.2
  • ...and 16 more