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Low-regularity invariant measure for the complex-valued mKdV

Zachary Lee, Nataša Pavlović, Gigliola Staffilani, Nicola Visciglia

Abstract

In this paper we consider the twice-renormalized, complex-valued modified KdV (mKdV) on the one-dimensional torus introduced by Chapouto. Our main result is the construction of an invariant measure supported at low-regularity. This work complements the work of Kenig et al., which constructed invariant measures supported in higher-regularity spaces for the non-renormalized mKdV. Due to the low-regularity of the support of the measure, we are forced to work in Fourier-Lebesgue spaces. The fact that we consider the complex-valued mKdV makes the problem more complicated than the real-valued case, which was previously considered.

Low-regularity invariant measure for the complex-valued mKdV

Abstract

In this paper we consider the twice-renormalized, complex-valued modified KdV (mKdV) on the one-dimensional torus introduced by Chapouto. Our main result is the construction of an invariant measure supported at low-regularity. This work complements the work of Kenig et al., which constructed invariant measures supported in higher-regularity spaces for the non-renormalized mKdV. Due to the low-regularity of the support of the measure, we are forced to work in Fourier-Lebesgue spaces. The fact that we consider the complex-valued mKdV makes the problem more complicated than the real-valued case, which was previously considered.
Paper Structure (23 sections, 25 theorems, 310 equations)

This paper contains 23 sections, 25 theorems, 310 equations.

Table of Contents

  1. Introduction
  2. Complex valued mKdV in the context of the $n$NLS hierarchy
  3. Renormalization of the mKdV
  4. Main Results of this Paper
  5. Difficulties and Novelties
  6. Notation
  7. Preliminaries
  8. Auxiliary spaces
  9. Decomposition of nonlinearity
  10. Multilinear estimates
  11. Definition of solution
  12. Finite-dimensional approximations
  13. Bounds uniform in $N$ for the flow $\Phi_N(t)u_0$
  14. Long-time convergence of $\Phi_{N}$
  15. Weighted Gaussian Measures, Pairing and Multilinear Gaussian estimates
  16. ...and 8 more sections

Key Result

Theorem 1.1

For every $(s,p)$ such that $1/2\le s <1-1/p, \quad 2<p<\infty$, and for both the focusing ($-$) and defocusing ($+$) signs there exists a Borel set $\Sigma^{s,p} \subset {\mathcal{F}} L^{s,p}(\mathbb{T})$ such that:

Theorems & Definitions (55)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • ...and 45 more