Ill-posedness in the critical Sobolev space for the $b$-Novikov equation

Technical Summary: Ill-posedness in the Critical Sobolev Space for the b-Novikov Equation

Problem Statement
The paper addresses the well-posedness of the Cauchy problem for the $b$-Novikov equation ($b$-NE), a one-parameter family of Camassa-Holm-type equations with cubic nonlinearities, defined on the real line $\mathbb{R}$. The equation is given by:
$$(1 - \partial_x^2)u_t = u^2 u_{xxx} + b u u_x u_{xx} - (b+1)u^2 u_x$$
While the equation is known to be locally well-posed (WP) in Sobolev spaces $H^s(\mathbb{R})$ for $s > 3/2$ and ill-posed (IP) for $s < 3/2$, the behavior at the critical index $s = 3/2$ remained unresolved for the general $b$-NE. Previous work by Guo et al. established ill-posedness for the specific case of the Novikov equation ($b=3$) and the $b$-family with quadratic nonlinearities, but the cubic case with $b \neq 3$ presented a significant obstacle due to the lack of known blow-up results and the absence of integrability for $b \neq 3$.

Methodology
The authors employ a proof by contradiction to demonstrate norm inflation in $H^{3/2}(\mathbb{R})$ for $b \geq 3$. The methodology proceeds through four main stages:

1. Refinement of Well-Posedness Estimates:
   The authors revisit the local well-posedness result for the generalized $k$-$b$ Camassa-Holm equation. They derive a sharper lower bound for the lifespan $T$ of solutions as the regularity index $s$ approaches the critical value $3/2$. Specifically, they show that the constant $c_s$ in the lifespan estimate behaves asymptotically as $c_s \sim \frac{1}{2s-3}$. This precise dependence is crucial for constructing initial data that exists long enough to exhibit the desired instability.

2. Derivation of a Crucial Differential Inequality:
   A central technical contribution is the derivation of a differential inequality for a nonlinear quantity evaluated along particle trajectories. By introducing the particle flow $q(x,t)$ and analyzing the evolution of $y = (1-\partial_x^2)u$, the authors establish that for initial data satisfying specific sign conditions ($y_0 \geq 0$ on $(-\infty, 0]$ and $y_0 \leq 0$ on $[0, \infty)$), the quantity $f(t) = (u u_x)(q(0,t), t)$ satisfies:
   $$\frac{d}{dt} f(t) \leq -\frac{b-2}{2} f^2(t) + \frac{b-2}{2} u^4(q(0,t), t)$$
   This inequality holds specifically when $b \geq 3$. Unlike previous approaches for the Novikov equation ($b=3$), which relied on integrability and conservation laws, this derivation relies solely on the structural properties of the equation and the sign preservation of the momentum density $y$.

3. Construction of Initial Data:
   The authors construct a sequence of initial data $u_{0,K}$ based on a sequence introduced by Guo et al. for the Novikov equation. The data is scaled by a small parameter $\epsilon$ and a large parameter $K$. The sequence is designed to:

   • Satisfy the sign conditions required for the differential inequality.
   • Have a small $H^{3/2}$ norm ($\sim \epsilon$).
   • Possess a large negative initial value for the product $u u_x$ at the origin ($\sim -\epsilon^2 K^{1/3}$).

4. Contradiction Argument:
   Assuming the equation is well-posed (specifically, that the data-to-solution map is continuous), the solution $v_K$ with scaled initial data should exist for a time $T > \epsilon$ with a bounded norm. However, by choosing $K$ and $s$ appropriately relative to $\epsilon$, the authors show that the differential inequality forces the quantity $f(t)$ to blow up (tend to $-\infty$) in finite time strictly less than $\epsilon$. This blow-up contradicts the assumption that the solution remains in $H^{3/2}$ (and thus the particle trajectory remains well-defined) up to time $\epsilon$.

Key Contributions and Results

• Main Theorem: The paper proves that the $b$-Novikov equation is ill-posed in the critical Sobolev space $H^{3/2}(\mathbb{R})$ for all $b \geq 3$.
• Norm Inflation: The ill-posedness is demonstrated via norm inflation. For every $\epsilon > 0$, there exists a solution $u_\epsilon$ and a time $t_\epsilon < \epsilon$ such that the initial norm is small ($\|u_\epsilon(0)\|_{H^{3/2}} \lesssim \epsilon$) while the norm at time $t_\epsilon$ is large ($\|u_\epsilon(t_\epsilon)\|_{H^{3/2}} \gtrsim \epsilon^{-1}$).
• Independence from Integrability: The proof provides an independent method for establishing ill-posedness for the Novikov equation ($b=3$) without relying on the conservation of the $H^1$ norm or other integrability features. This is significant because the method extends to non-integrable cases ($b > 3$) where such conservation laws are not available.
• Extension of Theory: The result completes the well-posedness theory for the $b$-Novikov equation, confirming that $s=3/2$ is the sharp threshold for local well-posedness.

Significance
The paper claims to resolve the endpoint case for the well-posedness of the $b$-Novikov equation. By overcoming the "fundamental obstacle" of the absence of blow-up results for non-integrable members of the family ($b \neq 3$), the authors establish that the critical Sobolev index $s=3/2$ belongs to the ill-posed regime. The work demonstrates that norm inflation occurs even in the absence of the special structural properties (like integrability) that facilitated previous proofs for the Novikov equation, thereby unifying the ill-posedness theory for this class of equations with cubic nonlinearities.