The zero-dispersion limit for the Benjamin--Ono equation on the circle

1. Problem Statement

The paper investigates the zero-dispersion limit of the Benjamin–Ono (BO) equation on the circle $\mathbb{T} = \mathbb{R}/2\pi\mathbb{Z}$. The BO equation models deep internal waves in stratified fluids and is a completely integrable system.

The problem considers a family of solutions $(u_\varepsilon)_{\varepsilon > 0}$ to the small-dispersion BO equation:
$$\partial_t u_\varepsilon + \partial_x \left( (u_\varepsilon)^2 - \varepsilon |D|u_\varepsilon \right) = 0, \quad u_\varepsilon(0, x) = u_0(x)$$
where $|D|$ is the Fourier multiplier with symbol $|\xi|$.

• The Limit: As the dispersion parameter $\varepsilon \to 0$, the equation formally approaches the inviscid Burgers equation $\partial_t u + \partial_x(u^2) = 0$.
• The Phenomenon: For $\varepsilon > 0$, shocks in the Burgers equation are replaced by rapid, high-frequency oscillations known as dispersive shocks. As $\varepsilon \to 0$, these oscillations become infinitely fast, preventing strong convergence (in $L^2$ or $L^\infty$).
• The Goal: To prove the existence of a weak limit $ZD[u_0]$ (the zero-dispersion limit) and to provide an explicit, computable characterization of this limit for bounded initial data $u_0 \in L^\infty(\mathbb{T}, \mathbb{R})$.

2. Methodology

The author employs a novel approach distinct from previous works on the line or for specific periodic data, leveraging recent breakthroughs in the integrability of the BO equation.

A. Explicit Formula and Scaling

The core of the methodology relies on the explicit formula for the BO solution discovered by Gérard [21].

1. Gérard's Formula: The solution $u(t)$ is determined by a function in the Hardy space $H^2(\mathbb{D})$ involving the inverse of an operator $(id - z e^{2it(D-T_{u_0})}S^*)^{-1}$, where $T_{u_0}$ is a Toeplitz operator and $S^*$ is the backward shift.
2. Scaling: The author scales this formula to the $\varepsilon$-dependent equation (BO-$\varepsilon$). By replacing $u_0$ with $u_0/\varepsilon$ and adjusting time, an explicit Fourier expansion for $u_\varepsilon$ is derived.
3. Weak Limit: Taking the limit $\varepsilon \to 0$ in the weak topology of $L^2(\mathbb{T})$, the author derives a series expansion for the limit function $ZD[u_0]$ in terms of powers of the Toeplitz operator $T_{u_0}$.

B. The Alternating Sum Characterization

The paper aims to show that this series expansion coincides with a "geometric" description involving the multivalued solution of the Burgers equation.

• Miller–Xu Formula: On the real line, Miller and Xu showed the limit is an alternating sum of the branches of the multivalued Burgers solution: $\sum (-1)^n u_B^n$.
• Generalization to the Circle: The author defines a generalized alternating sum formula using a kinetic formulation (the $\chi$-function):
  $$AS[u_0](t, x) = \int_{\mathbb{R}} \chi(u_0(x - 2ty), y) \, dy$$
  where $\chi(u, y) = \mathbb{1}_{u>y>0} - \mathbb{1}_{u<y<0}$. This integral represents the area between the graph of the multivalued solution and the $x$-axis, weighted by sign.

C. Combinatorial Proof of Equivalence

The most significant technical hurdle is proving that the operator-theoretic limit (from the explicit formula) equals the integral formula (from the Burgers solution).

• Series Expansion: Both expressions are expanded into power series in time $t$.
• Toeplitz Shifts: The coefficients of the operator expansion involve compositions of Toeplitz shifts ($S$ and $S^*$).
• Raney's Lemma: The author proves a combinatorial identity (Lemma 3.3), a variation of Raney's Lemma, to show that the complex weights arising from the Toeplitz operator compositions sum up exactly to the coefficients required by the integral formula. This step bridges the gap between the spectral/integrable approach and the hyperbolic/kinetic approach.

3. Key Contributions

1. Generalization to $L^\infty$ on the Circle:

   • Previous results on the circle (Gassot [19]) were restricted to "bell-shaped" data to utilize Birkhoff coordinates and eigenvalue asymptotics.
   • This paper extends the result to all bounded initial data $u_0 \in L^\infty(\mathbb{T}, \mathbb{R})$ using the explicit formula, which does not require knowledge of eigenvalues.
   • It also generalizes the line results of Chen [9] and Gérard [22] to the periodic setting.

2. Explicit Characterization:

   • The zero-dispersion limit is explicitly given by the integral formula $ZD[u_0](t, x) = \int_{\mathbb{R}} \chi(u_0(x - 2ty), y) \, dy$.
   • This confirms that the "alternating sum" structure observed on the line holds for periodic data, provided the sum is interpreted via the integral of the $\chi$-function (which handles the multivalued nature and caustics rigorously).

3. Combinatorial Insight:

   • The paper introduces a direct proof of a Raney-type lemma (Lemma 3.3) to equate the series expansions. This provides a new combinatorial tool for analyzing Toeplitz operators in the context of integrable PDEs.

4. Main Results

Theorem 1.1: Existence and Characterization

For any $u_0 \in L^\infty(\mathbb{T}, \mathbb{R})$, the solutions $u_\varepsilon$ converge weakly in $L^2(\mathbb{T})$ (locally uniformly in time) to a function $ZD[u_0]$. This limit is given by:
$$ZD[u_0](t, x) = \int_{\mathbb{R}} \left( \mathbb{1}_{u_0(x-2ty) > y > 0} - \mathbb{1}_{u_0(x-2ty) < y < 0} \right) dy$$
This formula generalizes the Miller–Xu result to the circle and arbitrary bounded data.

Theorem 1.2: Regularity and Properties

The mapping $(t, u_0) \mapsto ZD[u_0](t)$ possesses strong regularity properties analogous to entropy solutions of hyperbolic conservation laws:

• Maximum Principle: $\text{ess inf } u_0 \leq ZD[u_0](t) \leq \text{ess sup } u_0$.
• $L^p$-Contraction: $\|ZD[u_0](t) - ZD[v_0](t)\|_{L^1} \leq \|u_0 - v_0\|_{L^1}$.
• Norm Control: $\|ZD[u_0](t)\|_{L^p} \leq \|u_0\|_{L^p}$.
• Oleinik Estimate: A one-sided Lipschitz bound holds: $\frac{ZD[u_0](t, x) - ZD[u_0](t, y)}{x - y} \leq \frac{1}{2t}$.
• Strong Convergence: If $u_0 \in C^1$, convergence is strong ($L^2$) for times $t$ before the breaking time of the classical Burgers solution.
• Trotter Formula: The entropy solution of the Burgers equation can be recovered as the limit of iterated compositions of the zero-dispersion map.

5. Significance

• Unification of Approaches: The paper successfully unifies the spectral/integrable approach (using Birkhoff coordinates and explicit formulas) with the hyperbolic/kinetic approach (using Burgers characteristics and entropy solutions). It demonstrates that the zero-dispersion limit is a universal phenomenon for the BO equation, regardless of the domain (line vs. circle) or the specific regularity of the data (within $L^\infty$).
• Beyond Bell-Shaped Data: By removing the "bell-shaped" restriction required by previous periodic results, the paper provides a complete characterization for a much broader class of physical initial conditions.
• Methodological Innovation: The use of the explicit formula (Gérard) combined with a combinatorial lemma (Raney-type) offers a powerful new toolkit for studying zero-dispersion limits, potentially applicable to other integrable systems where explicit formulas are known but asymptotic analysis is difficult.
• Regularity Insights: The derivation of the Oleinik estimate and $L^1$-contraction for the zero-dispersion limit on the circle reinforces the deep connection between dispersive limits and the theory of entropy solutions for hyperbolic conservation laws.