On the singular nature of shallow-water convergence of the intermediate long wave equation on the real line

1. Problem Statement and Context

The paper investigates the Intermediate Long Wave (ILW) equation on the real line, which models internal waves at the interface of a stratified fluid of finite depth $\delta$. The equation is given by:
$$\partial_t u - G_\delta \partial_x^2 u = \partial_x(u^2)$$
where $G_\delta$ is a non-local operator depending on the depth parameter $\delta$.

The central problem addresses the shallow-water limit ($\delta \to 0$). To observe meaningful behavior in this limit, the equation is rescaled to:
$$\partial_t v - \tilde{G}_\delta \partial_x^2 v = \partial_x(v^2)$$
It is known that as $\delta \to 0$, this scaled ILW converges to the Korteweg-de Vries (KdV) equation:
$$\partial_t v + \partial_x^3 v = \partial_x(v^2)$$

The Paradox:

• The solution map for the KdV equation is known to be analytic in $H^s(\mathbb{R})$ for $s \geq -3/4$.
• However, for the ILW equation (and the Benjamin-Ono equation, its infinite-depth limit), the solution map is known to be not $C^2$-differentiable in $H^s(\mathbb{R})$ for any $s \in \mathbb{R}$ (a result by Molinet, Saut, and Tzvetkov).
• Question: How does the solution map gain regularity (from non-$C^2$ to analytic) in the shallow-water limit? The authors aim to explain the mechanism behind this "regularity gain" and the singular nature of the convergence.

2. Methodology

The authors employ a frequency decomposition strategy to separate the dynamics of the scaled ILW equation into two distinct components based on the depth parameter $\delta$.

A. Decomposition

Given a solution $v$ with initial data $\phi \in L^2(\mathbb{R})$, they decompose it as:
$$v = v_{\text{low}} + v_{\text{res}}$$
where:

• Low Frequency Part ($v_{\text{low}}$): Contains frequencies $|\xi| \lesssim \delta^{-1}$. It satisfies a modified equation where the nonlinearity is projected onto low frequencies.
• Residual Part ($v_{\text{res}}$): Contains high frequencies $|\xi| \gtrsim \delta^{-1}$. It satisfies an equation driven by the residual nonlinearity and interactions between low and high frequencies.

B. Analytical Tools

1. $X^{s,b}$ Spaces: The authors utilize Bourgain-type spaces adapted to the scaled ILW dispersion relation to establish local well-posedness and regularity estimates.
2. Resonance Function Analysis: A critical part of the methodology involves analyzing the resonance function $\tilde{\Xi}_\delta(\xi, \xi_1, \xi_2)$ for the scaled ILW.
   • For low frequencies ($|\xi| \lesssim \delta^{-1}$), the resonance behaves like the KdV resonance ($\sim \xi \xi_1 \xi_2$).
   • For high frequencies, it behaves differently, exhibiting a singularity as $\delta \to 0$.
3. Bilinear Estimates: They prove uniform bilinear estimates for the low-frequency dynamics that hold independently of $\delta$, allowing for a contraction mapping argument.
4. Gâteaux Derivatives: To prove the lack of regularity for the residual part, they compute the second-order Gâteaux derivative of the solution map at zero and construct specific initial data to show blow-up.

3. Key Contributions and Results

Theorem 1.1: Main Results

The paper establishes three main results regarding the solution map $S_\delta$ of the scaled ILW:

1. Analyticity of Low-Frequency Dynamics:
   For the low-frequency component $v_{\text{low}}$, the solution map $\phi \mapsto v_{\text{low}}$ is analytic in $H^s(\mathbb{R})$ for $s \geq 0$. Crucially, the time of existence and the analyticity bounds are uniform with respect to the depth parameter $\delta$ (for small $\delta$).

   • Significance: This explains why the "good" behavior of KdV is preserved in the low-frequency regime of ILW.

2. Failure of $C^2$-Regularity for Residual Dynamics:
   For the residual component $v_{\text{res}}$, the solution map is not $C^2$-differentiable at the zero function in $H^s(\mathbb{R})$ for any $s \in \mathbb{R}$.

   • Mechanism: The instability arises from the "low $\times$ high $\to$ high" frequency interaction. High frequencies (much larger than $\delta^{-1}$) are excited by the interaction of low and high frequencies. This specific interaction vanishes in the shallow-water limit ($\delta \to 0$) because the low-frequency cutoff $\delta^{-1}$ moves to infinity, effectively removing the "bad" interaction from the limit.

3. Shallow-Water Convergence:
   As $\delta \to 0$, the low-frequency solution $v_{\text{low}}$ converges to the solution of the KdV equation in $C(\mathbb{R}; H^s(\mathbb{R}))$.

   • Novelty: Unlike previous works that relied on the complete integrability of the ILW equation to prove convergence, this proof uses a semilinear contraction argument and establishes uniform equicontinuity of solutions without invoking integrability.

4. Technical Highlights

• Uniform Bilinear Estimates (Lemma 3.1): The authors prove that the bilinear estimate for the low-frequency part ($P_{\delta^{-1}} \partial_x (P_{\delta^{-1}} v_1)(P_{\delta^{-1}} v_2)$) holds uniformly in $\delta$. This relies on the fact that for $|\xi| \lesssim \delta^{-1}$, the dispersion relation of ILW approximates that of KdV ($\xi^3$), allowing the use of Kenig-Ponce-Vega type arguments.
• Resonance Function Bounds (Lemma 2.2): They rigorously quantify the transition in the resonance function:
  • If $\xi_{\max} \lesssim \delta^{-1}$, $|\tilde{\Xi}_\delta| \sim |\xi \xi_1 \xi_2|$ (KdV-like).
  • If $\xi_{\max} \gg \delta^{-1}$, $|\tilde{\Xi}_\delta| \sim \delta^{-1} \xi_{\min} \xi_{\max}$ (ILW-like, singular as $\delta \to 0$).
• Instability Construction (Section 4): To prove the failure of $C^2$ regularity, they construct initial data $\phi$ with a low-frequency component (scale $\alpha$) and a high-frequency component (scale $N$). The interaction generates a second derivative term that scales as $\delta^{1/2} N^{(1-\theta)/2}$. As $N \to \infty$, this term blows up, proving the map is not $C^2$.

5. Significance and Implications

• Resolution of the Regularity Paradox: The paper provides a rigorous explanation for why the solution map of ILW becomes analytic in the shallow-water limit despite being non-smooth for any fixed $\delta > 0$. The "gain" in regularity is not a smoothing of the equation itself, but a singular limit where the "bad" high-frequency interactions are pushed to infinity and disappear.
• Singular Nature of the Limit: The results highlight that the shallow-water limit is singular. The convergence is not uniform in the topology of the solution map's regularity; the "bad" part of the dynamics (residual) is lost in the limit, while the "good" part (low frequency) converges to KdV.
• Methodological Advancement: The proof of convergence (Theorem 1.1(iii)) is significant because it avoids the heavy machinery of complete integrability (Lax pairs, inverse scattering) used in previous convergence proofs (e.g., by the same authors in 2025). Instead, it relies on robust PDE techniques (contraction arguments and equicontinuity), suggesting these methods could be applied to other non-integrable dispersive equations with similar limits.
• Real Line vs. Periodic Case: The authors note that this phenomenon is specific to the real line. On the torus (periodic case), the "low $\times$ high $\to$ high" interaction does not cause the same instability for the Benjamin-Ono equation, and the solution map is analytic. This underscores the unique role of the continuous spectrum in the instability of the real-line ILW.

In summary, the paper decomposes the ILW equation to isolate the source of non-smoothness, demonstrating that the shallow-water limit is a singular process where the irregular high-frequency dynamics are excised, leaving behind the analytic low-frequency dynamics that converge to KdV.