Sharp quasi-invariance threshold for the cubic Szegő equation

Technical Summary of "Sharp Quasi-Invariance Threshold for the Cubic Szegő Equation"

Problem Statement
The paper investigates the statistical properties of the evolution of the one-dimensional cubic Szegő equation on the torus $\mathbb{T} = \mathbb{R}/(2\pi\mathbb{Z})$:
$$\begin{cases} i\partial_t u = \Pi(|u|^2 u), & (t, x) \in \mathbb{R} \times \mathbb{T}, \\ u(0) = u_0 = \Pi(u_0), \end{cases}$$
where $\Pi$ is the Szegő projection onto non-negative Fourier frequencies. The initial data $u_0$ is distributed according to a Gaussian measure $\mu_s$ with covariance operator $(1 - \partial_x^2)^{-s}$, where $s > 1/2$. The central question is whether the measure $\mu_s$ is quasi-invariant under the flow $\Phi_t$ of the equation. That is, does the transported measure $(\Phi_t)_\# \mu_s$ remain absolutely continuous with respect to the initial measure $\mu_s$ for all $t \in \mathbb{R}$?

The cubic Szegő equation serves as a "toy model" for non-dispersive Hamiltonian dynamics, specifically the zero-dispersion limit of fractional NLS or half-wave equations. Unlike dispersive PDEs (e.g., KdV, NLS) where quasi-invariance has been established for various Gaussian measures, the Szegő equation lacks dispersion and multilinear smoothing. This creates a tension: while conservation laws (mass $L^2$ and momentum $\dot{H}^{1/2}$) suggest potential invariance, the lack of dispersion resembles the behavior of the ODE $i\partial_t u = |u|^2 u$, where quasi-invariance is known to fail.

Methodology
The authors employ a combination of probabilistic analysis, harmonic analysis, and deterministic PDE techniques to establish a sharp threshold for quasi-invariance.

1. Heuristic and Variational Strategy (Quasi-invariance, $s > 1$):

   • The authors utilize a heuristic derived from the Liouville equation, suggesting that quasi-invariance depends on the finiteness of the expectation of the time-integrated divergence of the vector field, specifically the quantity $Q(u) = \frac{d}{dt}\|u\|_{H^s}^2$.
   • For $s > 1$, they prove that $Q(u)$ is square-integrable with respect to $\mu_s$.
   • The proof of quasi-invariance relies on a Boué-Dupuis variational formula to establish the exponential integrability of the density ratio. They adapt the strategy from previous works on dispersive equations (e.g., [28]) but refine it to handle the specific structure of the Szegő equation.
   • A key technical step involves a sharp Fourier truncation $\pi_N$ and proving uniform $L^p$ bounds on the truncated densities $f_{t,N}$, allowing the passage to the limit $N \to \infty$.

2. Singularity Strategy ($s < 1$):

   • For $s < 1$, the heuristic indicates that the relevant quantity diverges, suggesting singularity.
   • The authors develop a strategy to prove that for almost every time $t$, the evolved measure is mutually singular with the initial measure. This is achieved by showing that for a generic $u_0$, $t=0$ is a strict local minimum for the quantity $\|\Phi_t(u_0)\|_{H^s}^2$.
   • Paralinear Decomposition: To analyze the time evolution of the norm, the solution $u(t)$ is decomposed into $u(t) = X(t) + Y(t)$. Here, $X(t)$ captures the main nonlinear interaction (Low-Low-High) and retains the rough regularity of the initial data ($s-1/2$), while $Y(t)$ is a smoother remainder ($2s-1$).
   • Random Estimates: The authors compute the first and second time derivatives of the truncated $H^1$ norm (approximating the $H^s$ behavior). They show that the first derivative vanishes in expectation, while the second derivative scales as $(4s-3)N^{4-4s}$.
   • Crucially, they perform random estimates on the error terms. While deterministic bounds are insufficient to control the error terms relative to the leading order growth, the random oscillations of the initial data allow for sharper bounds (specifically, Lemma 4.14 provides a random estimate of order $N^{3-3s}$).
   • An abstract result (Proposition 4.1) is used to deduce that if the norm exhibits a local minimum behavior for almost every initial data, the measures must be singular for all but a countable set of times.

Key Contributions and Results
The main result, Theorem 1.1, establishes a sharp threshold for the regularity parameter $s$:

• Quasi-invariance ($s > 1$): If $s > 1$, the measure $\mu_s$ is quasi-invariant under the flow. Specifically, $(\Phi_t)_\# \mu_s \ll \mu_s$ for all $t \in \mathbb{R}$.
• Singularity ($1/2 < s < 1, s \neq 3/4$): If $1/2 < s < 1$ and $s \neq 3/4$, the transported measure $(\Phi_t)_\# \mu_s$ is mutually singular with respect to $\mu_s$ for almost every $t \in \mathbb{R}$. More precisely, there exists a countable set $N_s \subset \mathbb{R}$ such that for all $t \notin N_s$, $(\Phi_t)_\# \mu_s \perp \mu_s$.

Technical Nuances:

• The case $s=1$ is excluded from the singularity result because the leading order terms in the time derivative expansion are of the same order as the error terms, requiring sharper estimates not currently available.
• The case $s=3/4$ is excluded because the coefficient of the leading order term in the second derivative expansion vanishes (transitioning from positive to negative), making the sign of the growth indeterminate at this specific order.
• The space $X_s$ is carefully chosen to ensure local well-posedness and to contain the support of the Gaussian measure $\mu_s$. For $s \le 1$, this involves Besov spaces $B^{s-1/2}_{p,\infty}$ rather than standard Sobolev spaces.

Significance
The authors claim this work provides the first observation of a transition from quasi-invariance to singularity in the context of transporting Gaussian measures under the flow of Hamiltonian PDEs.

• Contrast with Dispersive Equations: In dispersive settings, quasi-invariance often holds even at low regularities due to smoothing effects. The Szegő equation, being non-dispersive, behaves more like an ODE, where singularity is expected. This paper rigorously quantifies the boundary between these behaviors.
• Sharpness: The result identifies $s=1$ as the critical threshold. Below this threshold (excluding specific points), the lack of dispersion leads to the breakdown of quasi-invariance, confirming the intuition that non-dispersive Hamiltonian dynamics can destroy the absolute continuity of Gaussian measures.
• Methodological Advancement: The paper introduces a flexible strategy for proving singularity that relies on the local minimization of the norm and random estimates for error terms, which the authors note was necessary for the work by Forlano and Tolomeo [13] and may be applicable to other non-dispersive models.

The paper concludes that the interplay between the conservation laws (which suggest invariance) and the lack of dispersion (which suggests singularity) results in a sharp phase transition at $s=1$, a phenomenon previously unobserved in this specific context.