Nonexistence of Hölder continuous solution for the Camassa-Holm equation in Besov spaces

1. Problem Statement

The paper addresses the regularity of the solution map for the classical Camassa-Holm (CH) equation:
$$\begin{cases} u_t - u_{xxt} + 3uu_x = 2u_x u_{xx} + u u_{xxx}, & (x, t) \in \mathbb{R} \times \mathbb{R}^+, \\ u(x, 0) = u_0(x), & x \in \mathbb{R}. \end{cases}$$
This equation is often rewritten in transport form as:
$$\partial_t u + u \partial_x u = P(u) := -\partial_x \Lambda^{-2} \left( u^2 + \frac{1}{2}(\partial_x u)^2 \right),$$
where $\Lambda^{-2} = (1-\partial_{xx})^{-1}$.

The Core Question:
While it is well-established that for initial data $u_0 \in B^s_{p,r}$ (with specific conditions on $s, p, r$), the CH equation admits a unique solution $u \in C([0, T]; B^s_{p,r})$, it remains an open question whether the solution map possesses higher temporal regularity. Specifically, the authors investigate if the solution belongs to the Hölder space $C^\alpha([0, T]; B^s_{p,r})$ for any $\alpha \in (0, 1)$.

The paper aims to prove that the solution map is not Hölder continuous in time, even when the initial data lies in the standard well-posedness regime.

2. Methodology

The authors employ a combination of Littlewood-Paley theory, Besov space estimates, and a constructive counterexample approach.

A. Functional Framework

The analysis is conducted in non-homogeneous Besov spaces $B^s_{p,r}(\mathbb{R})$. The authors utilize:

• Littlewood-Paley Decomposition: To decompose functions into dyadic frequency blocks $\Delta_j$.
• Bony's Paraproduct Decomposition: To handle nonlinear terms ($uv = T_u v + T_v u + R(u, v)$).
• Commutator Estimates: Specifically Lemma 2.4, which bounds commutators $[\Delta_j, v]\partial_x f$ to control the interaction between frequency localization and differentiation.

B. Construction of Initial Data

To disprove Hölder continuity, the authors construct a specific sequence of initial data $u_0$ that triggers "norm inflation" or rapid growth in the solution's deviation from the initial state.

• They define a smooth, compactly supported function $\phi$ and construct high-frequency oscillating functions:
  $$f_n(x) = \phi(x) \cos\left(\frac{17}{12} 2^n x\right).$$
• The initial data is defined as a series:
  $$u_0(x) = \sum_{n=3}^{\infty} n^{-2} 2^{-ns} \phi(x) \cos\left(\frac{17}{12} 2^n x\right).$$
• This construction ensures $u_0 \in B^s_{p,r}$ while concentrating energy at high frequencies ($2^n$) that interact strongly with the nonlinear transport term.

C. Error Estimation Strategy

The proof relies on analyzing the difference between the solution $S_t(u_0)$ and the initial data $u_0$.

1. First-order approximation: They define the error term $w(t) = S_t(u_0) - u_0 - t \tilde{e}_{u_0}$, where $\tilde{e}_{u_0} = P(u_0) - u_0 \partial_x u_0$.
2. Case Analysis:
   • Case 1 ($s > \max\{1+1/p, 3/2\}$): They show $\|w\|_{B^{s-2}_{p,r}} \lesssim t^2$.
   • Case 2 ($1+1/p < s \leq 3/2$ or critical endpoints): They show $\|w\|_{B^0_{p,r}} \lesssim t^s$.
3. Lower Bound Estimation: By isolating the dominant high-frequency term $\Delta_n$, they derive a lower bound for $\|S_t(u_0) - u_0\|_{B^s_{p,r}}$. The key insight is that the linear transport term $u_0 \partial_x u_0$ generates a term of order $2^n$ which dominates the error terms for small $t$.

3. Key Contributions

1. First Proof of Non-Hölder Continuity: This is the first work to rigorously demonstrate that the solution map for the classical CH equation fails to be Hölder continuous in time ($C^\alpha$) within the standard well-posedness Besov spaces.
2. Extension of Ill-posedness Results: As a by-product, the paper establishes the ill-posedness (discontinuity of the solution map at $t=0$) for the CH equation in $B^s_{p,\infty}(\mathbb{R})$ for $s > \max\{1+1/p, 3/2\}$. This improves upon previous results (e.g., Danchin, Guo-Liu-Molinet-Yin) by extending the range of regularity indices where ill-posedness holds, specifically covering the range $1+1/p < s \leq 3/2$.
3. Generalizability: The authors note that their techniques apply not only to the CH equation but also to the b-family equation and the Novikov equation, suggesting a broader lack of Hölder regularity for this class of integrable systems.

4. Main Results

Theorem 1.1 (Main Result)

Under the conditions:

• $s > 1 + 1/p$ with $(p, r) \in [1, \infty] \times [1, \infty)$, OR
• $s = 1 + 1/p$ with $(p, r) \in [1, \infty) \times \{1\}$,

For any $\alpha \in (0, 1)$, there exists an initial data $u_0 \in B^s_{p,r}(\mathbb{R})$ such that the solution map $u_0 \mapsto S_t(u_0)$ satisfies:
$$\limsup_{t \to 0^+} \frac{\|S_t(u_0) - u_0\|_{B^s_{p,r}}}{t^\alpha} = +\infty.$$
This implies that the solution $u(t)$ does not belong to $C^\alpha([0, T]; B^s_{p,r})$.

Corollary 1.1 (Ill-posedness in $B^s_{p,\infty}$)

For $s > 1 + 1/p$ and $p \in [1, \infty]$, there exists $u_0 \in B^s_{p,\infty}(\mathbb{R})$ such that the solution map is discontinuous at $t=0$:
$$\limsup_{t \to 0^+} \|S_t(u_0) - u_0\|_{B^s_{p,\infty}} \geq \varepsilon_0 > 0.$$

5. Significance

• Sharpness of Regularity: The results establish that the continuity of the solution map $C([0, T]; B^s_{p,r})$ is the optimal temporal regularity one can expect for the CH equation in these spaces. One cannot upgrade this to Hölder continuity without assuming higher regularity on the initial data (e.g., $u_0 \in B^{s'}_{p,r}$ with $s' > s$).
• Understanding Wave Breaking: The failure of Hölder continuity is closely linked to the "wave breaking" phenomenon inherent in the CH equation, where the slope becomes unbounded while the solution remains bounded. The lack of smoothness in the solution map reflects the sensitivity of the equation to high-frequency perturbations.
• Methodological Impact: The paper provides a refined framework for analyzing the continuity of solution maps for nonlinear dispersive and hyperbolic equations, utilizing subtle estimates in Besov spaces to handle critical regularity cases ($s \leq 3/2$) that were previously difficult to address.

In summary, the paper closes a significant gap in the mathematical theory of the Camassa-Holm equation by proving that while solutions exist and are continuous in time, they are inherently "rough" in the sense that they do not satisfy Hölder continuity, thereby defining the precise limits of the equation's well-posedness in Besov spaces.