A priori bounds and equicontinuity of orbits for the intermediate long wave equation

Technical Summary: A Priori Bounds and Equicontinuity of Orbits for the Intermediate Long Wave Equation

Problem Statement
The paper addresses the well-posedness and long-time behavior of solutions to the Intermediate Long Wave (ILW) equation, which models internal waves at the interface of a stratified fluid of depth $h > 0$. The equation is given by:
$$\frac{d}{dt}q = -Tq'' - \frac{1}{h}q' - 2qq'$$
where $T$ is a Fourier multiplier with symbol $i \coth(h\xi)$. The authors focus on the low-regularity regime $-1/2 < s \leq 0$ on both the real line ($M=\mathbb{R}$) and the circle ($M=\mathbb{T}$). While global well-posedness was recently established for $s > -1/2$ (specifically $s > -1/2$ for $M=\mathbb{T}$ and $L^2$ for $M=\mathbb{R}$), the behavior of solutions in the critical low-regularity range $-1/2 < s \leq 0$ regarding uniform-in-time bounds and the preservation of equicontinuity remained an open problem. Previous results in this range provided growth bounds that were exponential in time, dependent on the depth $h$.

Methodology
The authors' approach relies on the integrable structure of the ILW equation. The core of their methodology involves three main components:

1. Lax Pair Formulation: The authors identify a suitable Lax pair $(L, P)$ for the ILW equation, where $L(t) = -i\partial_x + \frac{1}{2h}(e^{2ih\partial_x} - 1) - q(t, x)$ and $P(t)$ is a specific differential operator. They verify that the ILW equation is equivalent to the Lax equation $\frac{d}{dt}L(t) = [P(t), L(t)]$. Crucially, they establish that the operator $L(t)$ is self-adjoint and bounded from below even for initial data $q \in H^s$ with $-1/2 < s \leq 0$.

2. Renormalized Perturbation Determinant: Mirroring techniques used for the Korteweg–de Vries (KdV) and Benjamin–Ono (BO) equations, the authors define a conserved quantity $\alpha(\kappa; q)$ based on the logarithm of the renormalized perturbation determinant. This is expressed as a series of traces:
   $$\alpha(\kappa; q) := \sum_{\ell=2}^{\infty} \frac{1}{\ell} \text{tr}\left\{ (\sqrt{R_0(\kappa)} q \sqrt{R_0(\kappa)})^\ell \right\}$$
   where $R_0(\kappa)$ is the resolvent of the free Lax operator. They prove that this quantity is conserved under the ILW flow.

3. Equicontinuity via Trace Estimates: The authors analyze the convergence and properties of the series defining $\alpha(\kappa; q)$. They utilize Hilbert-Schmidt estimates to bound the terms of the series. A key technical step involves analyzing the function $F(\xi; \kappa, h)$, which arises in the Hilbert-Schmidt norm of the resolvent-weighted potential. They demonstrate that for large $\kappa$, the series is dominated by its first term, which encodes the distribution of the solution across frequencies. This connection allows them to relate the conservation of $\alpha(\kappa; q)$ to the equicontinuity of the solution set in $H^s$.

Key Contributions and Results

• Uniform-in-Time A Priori Bounds (Theorem 1.1): The authors prove that for $-1/2 < s \leq 0$, the $H^s$ norm of smooth solutions to the ILW equation satisfies uniform-in-time bounds that are independent of the time $t$. Specifically, for any $h \geq h_0$:
  $$\|q(0)\|_{H^s}(1 + \|q(0)\|_{H^s})^{-2|s|} \lesssim \|q(t)\|_{H^s} \lesssim \|q(0)\|_{H^s}(1 + \|q(0)\|_{H^s})^{\frac{2|s|}{1-2|s|}}$$
  This improves upon previous results (e.g., Chapouto et al.) which exhibited exponential growth in time.

• Preservation of Equicontinuity: The paper proves that if a set of initial data $Q \subset H^\infty(M)$ is bounded and equicontinuous in $H^s$, then the set of orbits emanating from $Q$ under the ILW flow remains bounded and equicontinuous in $H^s$ uniformly for $h \geq h_0$. This is a significant step toward establishing global well-posedness in the low-regularity regime via compactness arguments.

• Shallow Water Limit (Theorem 1.2): The authors establish an analog of the main result for the shallow water limit ($h \to 0$). By rescaling the ILW equation to a form $(ILW')$ that formally converges to the KdV equation, they prove uniform-in-time bounds and equicontinuity preservation for solutions with initial data bounded in $H^s$ ($-1/2 < s < 0$) as $h \to 0$.

• Deep Water Limit Implications: The uniformity of the bounds and equicontinuity in the deep water limit ($h \to \infty$) provides strong evidence that the ILW equation converges locally in time to the Benjamin–Ono equation in $H^s(\mathbb{R})$ for all $s > -1/2$, extending known convergence results to lower regularities.

Significance
The paper claims that its primary significance lies in establishing the first uniform-in-time a priori bounds for the ILW equation in the low-regularity range $-1/2 < s \leq 0$, a regime where previous methods yielded only time-dependent growth. Furthermore, by proving the preservation of equicontinuity for sets of orbits, the authors provide the necessary compactness properties to treat the well-posedness problem in this low-regularity setting. The identification of a classical Lax pair formulation for ILW, which had previously been obscured by more complex zero-curvature representations, serves as the foundational tool enabling these results. The work bridges the gap between the known well-posedness for $s > -1/2$ and the ill-posedness results for $s < -1/2$, suggesting that the threshold $s = -1/2$ is sharp for the ILW equation in a manner consistent with the Benjamin–Ono equation.