Non-unique solutions to the periodic gKdV equation

Technical Summary: Non-Unique Solutions to the Periodic GKdV Equation

Problem Statement
This paper addresses the issue of unconditional uniqueness for the $k$-generalized Korteweg-de Vries ($k$-gKdV) equation on the torus $\mathbb{T}$:
$$\partial_t u + \partial_{xxx} u + \sigma \partial_x (u^k) = 0, \quad u(0, x) = u_0(x),$$
where $k \ge 2$ and $\sigma \in \{\pm 1\}$. While significant progress has been made in establishing well-posedness in Sobolev spaces $H^s(\mathbb{T})$ for various $s$, a critical gap remains regarding the behavior of solutions when the nonlinearity $u^k$ fails to be integrable (i.e., when $u \notin C^0_t L^k_x$).

In classical distribution theory, the term $\partial_x(u^k)$ requires $u^k$ to be a distribution, which typically necessitates $u \in L^k$. Below this threshold, the product $u^k$ is not well-defined in the standard sense. Previous works, such as those by Christ, have demonstrated non-uniqueness for KdV ($k=2$) in $C^0_t H^s_x$ for $s < 0$ by adopting a mollification-based definition of the nonlinearity. However, the regularity of these solutions is low, and the definition of the solution is relatively weak.

The central problem tackled here is to determine if non-uniqueness persists in "nearly optimal" regularity classes—specifically, just below the Lebesgue threshold $L^k_x$—and to formulate a notion of weak solution that is stronger than previous definitions yet capable of handling the singular nonlinearity.

Methodology
The authors employ a convex integration scheme adapted to the dispersive setting of the gKdV equation. The construction proceeds iteratively, generating a sequence of smooth approximate solutions $(u_q, E_q)$ that solve a relaxed equation:
$$\partial_t u_q + \partial_{xxx} u_q + \sigma \partial_x (u_q^k) = \partial_x E_q,$$
where the error term $E_q$ is driven to zero in a specific norm.

Key technical components include:

1. Intermittent Building Blocks: The perturbations $w_{q+1}$ added at each step are constructed using "intermittent slabs" (Lemma 2.11). These are highly concentrated spatial profiles that are $L^k$-normalized but small in subcritical $L^p$ norms ($p < k$). This intermittency allows the $k$-th power of the perturbation to cancel the existing error while keeping lower-order norms summable.
2. Frequency Localization: The perturbations are frequency-localized to scales $\lambda_{q+1}$ with specific intermittency parameters $\epsilon_{q+1}$. This ensures that the Sobolev norms of the increments are summable for regularities $\alpha < \frac{1}{2} - \frac{1}{k}$, while maintaining sufficient $L^k$ strength to cancel errors.
3. Weighted Wiener Spaces: A novel aspect of the methodology is the use of homogeneous weighted Wiener spaces ($\dot{A}^s$) to measure errors. Instead of relying solely on Sobolev norms, the authors measure the absolute summability of the Fourier coefficients of the nonlinear terms. This allows for precise control over the "oscillation error" generated by the high-frequency perturbations.
4. Error Decomposition: The error $E_{q+1}$ is decomposed into four parts:
   • Oscillation Error ($E_O$): Canceled by the mean of the $k$-th power of the perturbation.
   • Nash Error ($E_N$): Arising from lower-order cross-terms, controlled by the smallness of the perturbation in $L^p$ for $p < k$.
   • Dispersion Error ($E_D$): Controlled by the $C^0_t L^1_x$ norm of the perturbation.
   • Temporal Error ($E_T$): Controlled by the time derivative of the perturbation's antiderivative.

Key Contributions and Definitions
The paper introduces a new framework for interpreting the nonlinearity in regimes where it is not classically integrable:

• Definition of Weak Singular Solutions (Definition 1.3): A solution $u$ is defined as a "weak singular solution" if the nonlinearity $P_{\neq 0}(u^k)$ (the non-zero frequency modes of $u^k$) is well-defined as an element of $C^0_t \dot{H}^{-s}_x$ via absolute Fourier summability (Definition 1.2). Specifically, the series $\sum_{\xi_1 + \dots + \xi_k \neq 0} |\hat{u}(\xi_1) \dots \hat{u}(\xi_k)| |\xi_1 + \dots + \xi_k|^{-s}$ must converge.
• Comparison to Previous Notions: The authors prove (in the Appendix) that this notion is strictly stronger than:
  • The "paraproduct" notion used in previous convex integration works for dispersive equations.
  • Christ's mollification-based notion of solution.
    Consequently, the constructed solutions are valid in these broader frameworks as well.

Main Results
The primary result is Theorem 1.4, which establishes the existence of non-trivial weak singular solutions with identically zero initial data ($u(0, x) = 0$) in the following regularity classes:

1. For $k=2$ (KdV): There exists a non-zero solution $u \in \bigcap_{\epsilon > 0} C^0_t L^{2-\epsilon}_x([0, 1] \times \mathbb{T})$. This result is sharp in the Lebesgue scale, as unconditional uniqueness is known to hold in $C^0_t L^2_x$ (referencing Babin-Ilyin-Titi [2]).
2. For $k \ge 3$: There exists a non-zero solution $u \in \bigcap_{\epsilon > 0} C^0_t H^{\frac{1}{2} - \frac{1}{k} - \epsilon}_x([0, 1] \times \mathbb{T})$. This reaches the Sobolev endpoint-minus scale corresponding to the Lebesgue threshold $L^k_x$.

Additionally, Remark 1.5 notes that for $k=2$, the solution can be refined to lie in $\bigcap_{\epsilon > 0} C^0_t B^{-s}_{\infty, \infty, x}$, making it "nearly continuous" as well as "nearly square integrable."

Significance and Conjectures
The paper argues that the failure of the convex integration scheme to close when the nonlinearity lies in $C^0_t L^1_x$ (i.e., when $u \in C^0_t L^k_x$) suggests a fundamental threshold for uniqueness.

• Necessary Condition: The result implies that a necessary condition for unconditional uniqueness for $k$-gKdV is that the nonlinearity lies in $C^0_t L^1_x$ (which corresponds to $u \in C^0_t L^k_x$).
• Conjecture 1.1: Based on the sharpness of the KdV result and the rigidity of the construction below the threshold, the authors conjecture that the $k$-gKdV equation is unconditionally locally well-posed in $H^s(\mathbb{T})$ for all $s \ge \frac{1}{2} - \frac{1}{k}$.

Extensions
The methodology is shown to be robust and applicable to other dispersive equations:

• Stationary gKdV: The authors extend the result to the stationary equation, proving both rigidity (smoothness of $L^k$ solutions) and flexibility (non-uniqueness below $L^k$).
• Generalized Benjamin-Ono (gBO) and Other Dispersive Models: The scheme applies to equations with linear operators $L$ where the symbol can be divided by $\partial_x$ with polynomial loss (e.g., gBO, higher-order dispersive models). The dispersion error estimates are often easier for these operators than for the standard KdV dispersion.

In summary, this work utilizes a refined convex integration scheme with intermittent building blocks and weighted Wiener norms to construct non-unique solutions to the periodic gKdV equation in regularity classes just below the threshold where the nonlinearity becomes classically integrable, thereby identifying a sharp boundary for unconditional uniqueness.