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Qualitative quasi-invariance of low regularity Gaussian measures for the 1d quintic nonlinear Schrödinger equation

Alexis Knezevitch

Abstract

We consider the 1d quintic nonlinear Schrödinger equation (NLS) on the torus with initial data distributed according to the Gaussian measures with covariance operator $(1-Δ)^{-s}$, and denoted $μ_s$. For the full range $s>\frac{9}{10}$, we prove that these Gaussian measures are quasi-invariant along the flow of (NLS), meaning that the law of the solution at any time is absolutely continuous with respect to the initial Gaussian measure. Moreover, the condition $s>\frac{9}{10}$ corresponds to the threshold where the Sobolev space $H^{\frac{2}{5}+}(\mathbb{T})$ is of $μ_s$-full measure (it is of zero $μ_s$-measure otherwise). This is the lower regularity Sobolev space where we currently know that (NLS) is globally well-posed, thanks to a work by LI-WU-XU. The present work extends the known threshold $s>\frac{3}{2}$ for the quasi-invariance down to $s>\frac{9}{10}$, but we do not obtain here quantitative results on the Radon-Nikodym derivatives. Our approach is based on a work of Sun-Tzvetkov, combining a Poincaré-Dulac normal form reduction with energy estimates. However, our main tool to obtain these energy estimates differs: we use the Boué-Dupuis variational formula instead of Wiener Chaos.

Qualitative quasi-invariance of low regularity Gaussian measures for the 1d quintic nonlinear Schrödinger equation

Abstract

We consider the 1d quintic nonlinear Schrödinger equation (NLS) on the torus with initial data distributed according to the Gaussian measures with covariance operator , and denoted . For the full range , we prove that these Gaussian measures are quasi-invariant along the flow of (NLS), meaning that the law of the solution at any time is absolutely continuous with respect to the initial Gaussian measure. Moreover, the condition corresponds to the threshold where the Sobolev space is of -full measure (it is of zero -measure otherwise). This is the lower regularity Sobolev space where we currently know that (NLS) is globally well-posed, thanks to a work by LI-WU-XU. The present work extends the known threshold for the quasi-invariance down to , but we do not obtain here quantitative results on the Radon-Nikodym derivatives. Our approach is based on a work of Sun-Tzvetkov, combining a Poincaré-Dulac normal form reduction with energy estimates. However, our main tool to obtain these energy estimates differs: we use the Boué-Dupuis variational formula instead of Wiener Chaos.

Paper Structure

This paper contains 22 sections, 31 theorems, 277 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. The resonant function and the symmetrized derivatives
  5. Probabilistic toolbox
  6. Poincaré-Dulac normal form reduction and energy estimates
  7. Truncated system
  8. Poincaré-Dulac normal form reduction
  9. Exponential integrability and quantitative TEXT estimates
  10. Transport of Gaussian measures by the truncated flow
  11. Proof of the quasi-invariance
  12. Lower bound on the resonant function and counting bounds
  13. A lower bound on the resonant function
  14. Counting bounds
  15. Deterministic estimates
  16. ...and 7 more sections

Key Result

Theorem 1.1

For every $s > \frac{9}{10}$, the Gaussian measure $\mu_s$ is quasi-invariant along the flow of NLS. In other words, the push-forward measure $\Phi(t)_\# \mu_s$ is absolutely continuous with respect to $\mu_s$ for every time $t\in \mathbb{R}$.

Theorems & Definitions (68)

  • Theorem 1.1
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5: Factorization of the degenerated resonant function
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 58 more