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A continuum of invariant measures for the periodic KdV and mKdV equations

Andreia Chapouto, Justin Forlano

Abstract

We consider the real-valued defocusing modified Korteweg-de Vries equation (mKdV) on the circle. Based on the complete integrability of mKdV, Killip-Vişan-Zhang (2018) discovered a conserved quantity which they used to prove low regularity a priori bounds for solutions. It has been an open question if this conserved quantity can be used to define invariant measures supported at fractional Sobolev regularities. Motivated by this question, we construct probability measures supported on $H^s(\mathbb{T})$ for $0<s<1/2$ invariant under the mKdV flow. We then use the Miura transform to obtain invariant measures for the Korteweg-de Vries equation, whose supports are rougher than the white noise measure. We also obtain analogous results for the defocusing cubic nonlinear Schrödinger equation. These invariant measures cover the lowest possible regularities for which the flows of these equations are well-posed.

A continuum of invariant measures for the periodic KdV and mKdV equations

Abstract

We consider the real-valued defocusing modified Korteweg-de Vries equation (mKdV) on the circle. Based on the complete integrability of mKdV, Killip-Vişan-Zhang (2018) discovered a conserved quantity which they used to prove low regularity a priori bounds for solutions. It has been an open question if this conserved quantity can be used to define invariant measures supported at fractional Sobolev regularities. Motivated by this question, we construct probability measures supported on for invariant under the mKdV flow. We then use the Miura transform to obtain invariant measures for the Korteweg-de Vries equation, whose supports are rougher than the white noise measure. We also obtain analogous results for the defocusing cubic nonlinear Schrödinger equation. These invariant measures cover the lowest possible regularities for which the flows of these equations are well-posed.
Paper Structure (11 sections, 26 theorems, 218 equations)

This paper contains 11 sections, 26 theorems, 218 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Structures for mKdV
  5. Construction of the measures
  6. Construction of the measures
  7. Equivalence of base Gaussian measures
  8. Invariance
  9. Proof of invariance under $H_{\varkappa}^{\text{mKdV}}$ and $H^{\text{mKdV}}$
  10. Invariant measures for KdV
  11. On the defocusing cubic NLS

Key Result

Theorem 1.1

Let $\tfrac{1}{2}<s<1$ and $R>0$. Then, $\rho_{s,R}$ in GibbsEmeas defines a probability measure on $L^2(\mathbb{T})$, endowed with the Borel sigma algebra, which satisfies: (i)$\rho_{s,R}$ is absolutely continuous with respect to the Gaussian measure $\mu_{s}$ in gauss0. (ii)$\mathop{\mathrm{supp}} (iii) The measure $\rho_{s,R}$ is invariant under the defocusing mKdV flow mkdv. More precisely, fo

Theorems & Definitions (52)

  • Theorem 1.1: Invariant measures for defocusing mKdV
  • Corollary 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Lemma 2.1
  • ...and 42 more