Nonlinear PDEs with modulated dispersion II: Korteweg-de Vries equation

Technical Summary: Nonlinear PDEs with Modulated Dispersion II: Korteweg-de Vries Equation

Problem Statement
This paper investigates dispersive partial differential equations (PDEs) featuring a time non-homogeneous modulation acting on the linear dispersion term. The primary model is the modulated Korteweg-de Vries (KdV) equation on both the circle ($\mathbb{T}$) and the real line ($\mathbb{R}$):
$$\partial_t u + \partial_x^3 u \cdot \partial_t w = \partial_x (u^2),$$
where $w: \mathbb{R}_+ \to \mathbb{R}$ is an arbitrary continuous function representing the modulation. The authors also study related equations, including the modulated modified KdV (mKdV), Benjamin-Ono (BO), and Intermediate Long Wave (ILW) equations.

A central challenge is that the modulation $w$ is often irregular (e.g., sample paths of Brownian motion or fractional Brownian motion), meaning the time derivative $\partial_t w$ does not exist in the classical sense. Consequently, standard analytical tools like the Fourier restriction norm method (Bourgain's method) are not directly applicable. The paper aims to establish well-posedness results for these equations in regimes where the unmodulated counterparts are known to be ill-posed or exhibit quasilinear behavior, specifically in negative Sobolev spaces.

Methodology
The authors employ a pathwise approach based on nonlinear Young integration, a technique developed in their previous work on modulated nonlinear Schrödinger (NLS) equations and rooted in the theory of controlled rough paths (Gubinelli, Lyons).

1. Duhamel Formulation and Interaction Representation: Instead of interpreting the equation as a stochastic PDE (which requires $w$ to be a semimartingale), the authors work with the mild (Duhamel) formulation. They introduce a modulated interaction representation $\mathbf{u}(t) = U_w(t)^{-1} u(t)$, where $U_w(t) = e^{-w(t)L}$ is the modulated linear propagator. This transforms the equation into a fixed-point problem:
   $$\mathbf{u}(t) = u_0 + \int_0^t U_w(t')^{-1} N(U_w(t') \mathbf{u}(t')) \, dt'.$$
2. Nonlinear Young Integrals: The integral term is interpreted as a nonlinear Young integral. This requires the modulation $w$ to satisfy a specific irregularity condition (Definition 1.1), quantified by the norm $\|\Phi_w\|_{W^{\rho, \gamma}_T}$. This condition essentially measures the oscillation of the modulation in the frequency domain.
3. Sewing Lemma: The construction of the integral relies on the sewing lemma (Lemma 3.3), which allows for the definition of integrals against paths with low temporal regularity, provided the sum of the path's regularity and the driver's regularity exceeds 1.
4. I-Method (Almost Conservation Laws): For global well-posedness in negative Sobolev spaces, the authors adapt the I-method. They combine a crucial commutator estimate with the sewing lemma to control the growth of a modified energy functional.
5. Probabilistic Tools: For specific applications involving random initial data (white noise) or additive space-time white noise, the authors utilize Bourgain's invariant measure argument and evolution systems of measures.

Key Contributions and Results

The paper establishes three distinct forms of regularization by noise, where the irregularity of the modulation $w$ improves the well-posedness properties of the equations:

1. Well-posedness in Ill-posed Regimes:

   • Modulated KdV on $\mathbb{T}$: For any $s \in \mathbb{R}$, if the modulation is sufficiently irregular (specifically, if $w$ is $(\rho, \gamma)$-irregular with $\rho > 1/2$ and appropriate $\gamma$), the modulated KdV is locally well-posed in $H^s(\mathbb{T})$. This contrasts with the unmodulated KdV, which is ill-posed for $s < -1/2$ on the circle.
   • Modulated KdV on $\mathbb{R}$: The equation is locally well-posed in $H^s(\mathbb{R})$ for $s > -3/2$ (the scaling-critical threshold), even though the unmodulated KdV is ill-posed for $s < -3/4$.
   • Corollary: If the modulation is a fractional Brownian motion with Hurst index $0 < H < 2/3$, the modulated KdV is locally well-posed in $H^s(\mathbb{T})$ for $s > -1/(2H)$.

2. Semilinearization:

   • It is known that unmodulated dispersive equations like KdV and BO exhibit quasilinear nature (failure of uniform continuity of the solution map) below certain regularity thresholds.
   • The authors prove that sufficiently irregular modulations render these equations semilinear. The solution map becomes locally Lipschitz continuous, established via a contraction argument in the space of paths controlled by $w$. This holds for KdV, mKdV, BO, and ILW on the circle.

3. Nonlinear Smoothing:

   • The paper proves that the nonlinear part of the solution gains regularity. Crucially, this gain becomes arbitrarily large as the modulation becomes more irregular. This contrasts with the unmodulated case where the gain is fixed.

Specific Theorems and Applications:

• Theorem 1.3 & 1.7: Establish local and global well-posedness for modulated KdV on $\mathbb{T}$ and $\mathbb{R}$ in negative Sobolev spaces under irregular modulation conditions.
• Theorem 1.10: Addresses the modulated mKdV, showing local well-posedness for $s \ge 1/2$. The authors note that unlike KdV, the modulation does not improve the regularity threshold for mKdV due to resonant interactions where the modulation plays no role.
• Theorem 1.12: Proves local and global well-posedness for modulated BO and ILW on $\mathbb{T}$ in $H^s$ for any $s \in \mathbb{R}$ (given sufficient irregularity), establishing a strong regularization effect and semilinearization in negative Sobolev spaces.
• Theorem 1.14: Establishes deep-water ($\delta \to \infty$) and shallow-water ($\delta \to 0$) convergence for the modulated ILW to the modulated BO and KdV, respectively, in arbitrary negative Sobolev spaces.
• Theorem 1.20 & 1.23:
  • Proves that the white noise measure is invariant under the dynamics of the modulated KdV on the circle.
  • Establishes global well-posedness for the stochastic modulated KdV with additive space-time white noise and white noise initial data. The family of measures $\{\mu_{1+t}\}_{t \ge 0}$ forms an evolution system of measures.
• Theorem 1.21: Extends the invariance of white noise and global well-posedness to the modulated BO and ILW equations.

Significance and Claims
The paper claims to provide the first results on regularization by noise for nonlinear dispersive equations with rough (deterministic) initial data. By developing a pathwise approach using nonlinear Young integration, the authors bypass the need for stochastic calculus interpretations (Itô/Stratonovich) that require specific probabilistic structures on the modulation.

The authors emphasize that their results are deterministic; the stochastic applications (e.g., using fractional Brownian motion as a modulation) are consequences of the pathwise analysis. They highlight that the "semilinearization" phenomenon—where irregular modulation turns a quasilinear equation into a semilinear one—is a novel and strong effect observed in this setting.

The work also introduces a novel combination of tools: the I-method (typically used for global well-posedness in dispersive PDEs) is successfully adapted to the modulated setting by integrating it with the sewing lemma from rough path theory. This allows for global well-posedness in negative Sobolev spaces for the modulated KdV.

Finally, the paper notes that while the results are limited to sufficiently irregular modulations, they do not yet provide a sharp characterization of the transition between smooth and irregular modulations, leaving the gap between regular and irregular regimes as an open problem.