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Invariant Gibbs dynamics for two-dimensional fractional wave equations in negative Sobolev spaces

Luigi Forcella, Oana Pocovnicu

TL;DR

This work studies the defocusing two-dimensional fractional nonlinear wave equation $u_{tt}+(1-\Delta)^\alpha u+u^{2m+1}=0$ on the torus, aiming to construct a renormalized Gibbs measure and to prove almost sure global well-posedness for Gibbs-random initial data in negative Sobolev spaces. The authors implement Wick renormalization of the energy and employ Barashkov–Gubinelli’s variational method to obtain exponential integrability, enabling a well-posed Gibbs dynamics framework. Solutions are constructed via a two-tier expansion: a first order Da Prato–Debussche decomposition $u=z+w$ and, for enhanced regularity, a second order expansion $u=z+z_2+w_2$, with precise α-ranges that depend on the nonlinearity degree $m$. The main contributions include a generalized construction of renormalized Gibbs measures for odd power nonlinearities, an almost sure local well-posedness theory in negative Sobolev spaces, and global-in-time invariance results obtained by Bourgain’s invariant measure and approximation argument, thereby extending invariant Gibbs dynamics to fractional wave equations in low regularity settings.

Abstract

We consider a fractional nonlinear wave equations (fNLW) with a general power-type nonlinearity, on the two-dimensional torus. Our main goal is to construct invariant global-in-time Gibbs dynamics for a renormalized fNLW. We first construct the Gibbs measure associated with this equation by using the variational approach of Barashkov and Gubinelli. We then prove almost sure local well-posedness with respect to Gibbsian initial data, by exploiting the second order expansion. Finally, we extend solutions globally in time using Bourgain's invariant measure argument.

Invariant Gibbs dynamics for two-dimensional fractional wave equations in negative Sobolev spaces

TL;DR

This work studies the defocusing two-dimensional fractional nonlinear wave equation on the torus, aiming to construct a renormalized Gibbs measure and to prove almost sure global well-posedness for Gibbs-random initial data in negative Sobolev spaces. The authors implement Wick renormalization of the energy and employ Barashkov–Gubinelli’s variational method to obtain exponential integrability, enabling a well-posed Gibbs dynamics framework. Solutions are constructed via a two-tier expansion: a first order Da Prato–Debussche decomposition and, for enhanced regularity, a second order expansion , with precise α-ranges that depend on the nonlinearity degree . The main contributions include a generalized construction of renormalized Gibbs measures for odd power nonlinearities, an almost sure local well-posedness theory in negative Sobolev spaces, and global-in-time invariance results obtained by Bourgain’s invariant measure and approximation argument, thereby extending invariant Gibbs dynamics to fractional wave equations in low regularity settings.

Abstract

We consider a fractional nonlinear wave equations (fNLW) with a general power-type nonlinearity, on the two-dimensional torus. Our main goal is to construct invariant global-in-time Gibbs dynamics for a renormalized fNLW. We first construct the Gibbs measure associated with this equation by using the variational approach of Barashkov and Gubinelli. We then prove almost sure local well-posedness with respect to Gibbsian initial data, by exploiting the second order expansion. Finally, we extend solutions globally in time using Bourgain's invariant measure argument.
Paper Structure (16 sections, 19 theorems, 141 equations)

This paper contains 16 sections, 19 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Renormalized Gibbs measures
  3. Solution theory for renormalized fNLW
  4. First order expansion
  5. Second order expansion
  6. Organisation of the paper
  7. Preliminary deterministic and stochastic tools
  8. Deterministic estimates
  9. Regularity of stochastic objects
  10. Construction of the renormalized Gibbs measures
  11. Convergence of random variables
  12. Exponential integrability
  13. Proof of Theorem \ref{['thm:exp:int']}
  14. First order expansion almost sure local well-posedness
  15. Second order expansion almost sure local well-posedness
  16. ...and 1 more sections

Key Result

Proposition 1.1

Let $m \in \mathbb{N}$ and let $\alpha\in\left(1- \frac{1}{2m+2},1\right)$. Let $1\leq p\leq \infty$. Let $u=u^\omega$ be a typical element under the measure $\mu_{\alpha}$ defined in G4. The sequence $\{G_N(u)\}_{N\in\mathbb{N}}$ is a Cauchy sequence in $L^p(\mu_{\alpha})$. More specifically, there We therefore define

Theorems & Definitions (42)

  • Proposition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 32 more