Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantitative quasi-invariance of Gaussian measures below the energy level for the 1D generalized nonlinear Schrödinger equation and application to global well-posedness

Alexis Knezevitch

TL;DR

This work proves almost-sure global well-posedness for the 1D periodic nonlinear Schrödinger equation with odd nonlinearities $p\ge 5$ for rough Gaussian initial data below the energy level. The authors blend a deterministic local Cauchy theory with a probabilistic globalization that leverages the quasi-invariance of Gaussian measures $\mu_s$ under the flow, together with $L^q$ bounds on Radon-Nikodym derivatives. Central to the approach are the renormalized energy and a Boué-Dupuis variational framework, augmented by a Poincaré-Dulac normal form reduction to control resonant interactions. The result yields a global flow for $\mu_s$-typical initial data with polynomial-in-time Sobolev-norm growth and establishes a robust probabilistic globalization parallel to Bourgain’s invariant-measure strategy, informed by recent developments on transport and quasi-invariance. The techniques provide a template for combining deterministic and probabilistic methods to achieve global behavior for dispersive PDEs with rough random data.

Abstract

We consider the Schrödinger equation on the one dimensional torus with a general odd-power nonlinearity $p \geq 5$, which is known to be globally well-posed in the Sobolev space $H^σ(\mathbb{T})$, for every $σ\geq 1$, thanks to the conservation and finiteness of the energy. For regularities $σ< 1$, where this energy is infinite, we explore a globalization argument adapted to random initial data distributed according to the Gaussian measures $μ_s$, with covariance operator $(1-Δ)^s$, for $s$ in a range $(s_p,\frac{3}{2}]$. We combine a deterministic local Cauchy theory with the quasi-invariance of Gaussian measures $μ_s$, with additional $L^q$-bounds on the Radon-Nikodym derivatives, to prove that the Gaussian initial data generate almost surely global solutions. These $L^q$-bounds are obtained with respect to Gaussian measures accompanied by a cutoff on a renormalization of the energy; the main tools to prove them are the Boué-Dupuis variational formula and a Poincaré-Dulac normal form reduction. This approach is similar in spirit to Bourgain's invariant argument and to a recent work by Forlano-Tolomeo.

Quantitative quasi-invariance of Gaussian measures below the energy level for the 1D generalized nonlinear Schrödinger equation and application to global well-posedness

TL;DR

This work proves almost-sure global well-posedness for the 1D periodic nonlinear Schrödinger equation with odd nonlinearities for rough Gaussian initial data below the energy level. The authors blend a deterministic local Cauchy theory with a probabilistic globalization that leverages the quasi-invariance of Gaussian measures under the flow, together with bounds on Radon-Nikodym derivatives. Central to the approach are the renormalized energy and a Boué-Dupuis variational framework, augmented by a Poincaré-Dulac normal form reduction to control resonant interactions. The result yields a global flow for -typical initial data with polynomial-in-time Sobolev-norm growth and establishes a robust probabilistic globalization parallel to Bourgain’s invariant-measure strategy, informed by recent developments on transport and quasi-invariance. The techniques provide a template for combining deterministic and probabilistic methods to achieve global behavior for dispersive PDEs with rough random data.

Abstract

We consider the Schrödinger equation on the one dimensional torus with a general odd-power nonlinearity , which is known to be globally well-posed in the Sobolev space , for every , thanks to the conservation and finiteness of the energy. For regularities , where this energy is infinite, we explore a globalization argument adapted to random initial data distributed according to the Gaussian measures , with covariance operator , for in a range . We combine a deterministic local Cauchy theory with the quasi-invariance of Gaussian measures , with additional -bounds on the Radon-Nikodym derivatives, to prove that the Gaussian initial data generate almost surely global solutions. These -bounds are obtained with respect to Gaussian measures accompanied by a cutoff on a renormalization of the energy; the main tools to prove them are the Boué-Dupuis variational formula and a Poincaré-Dulac normal form reduction. This approach is similar in spirit to Bourgain's invariant argument and to a recent work by Forlano-Tolomeo.

Paper Structure

This paper contains 20 sections, 25 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The resonant function and the symmetrized derivatives
  4. Probabilistic tools
  5. Quantitative quasi-invariance and globalization of the local solutions
  6. Local Cauchy theory and approximation
  7. Transport of cutoff Gaussian measures
  8. The globalization argument
  9. Quantitative quasi-invariance for the original flow and application
  10. Proof of the quasi-invariance
  11. Growth of Sobolev norms
  12. Renormalized energy and associated cutoff Gaussian measures
  13. Estimates with bounded renormalized energy
  14. Poincaré-Dulac normal form reduction and deterministic estimate
  15. Poincaré-Dulac normal form reduction
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $p \geq 5$ an odd integer. Let $\frac{3}{2} \geq s>s_p$, where $s_p$ is defined by: Then, for $\sigma < s - \frac{1}{2}$ close to $s-\frac{1}{2}$, we have

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2: Quantitative quasi-invariance
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.4
  • Lemma 2.5
  • Remark 2.6
  • Lemma 2.7
  • ...and 37 more