周期Camassa–Holm方程在临界Triebel–Lizorkin空间中的尖锐正则性
Sharp regularity for the periodic Camassa--Holm equation in critical Triebel--Lizorkin spaces
AI总结：
该研究针对周期Camassa–Holm方程，在临界Triebel–Lizorkin空间中建立了尖锐适定性与范数膨胀理论，补充了临界Besov空间等的局部适定性，为相关方程的正则性研究提供了关键结果。
AI中文摘要：
我们在临界Triebel–Lizorkin空间$F^{1+1/p}_{p,q}(\mathbb{T})$中建立了Camassa–Holm方程的尖锐适定性与范数膨胀理论。在端点$p=1$处，我们证明了$1\le q<\infty$时的局部Hadamard适定性；与之相反，证明了$1<p<\infty$且$1\le q\le\infty$时的范数膨胀。我们还补充了临界Besov空间及更高正则性Triebel–Lizorkin空间中的局部适定性。正结果依赖于周期Green算子在一度拉格朗日流下的Lipschitz稳定性定理；负结果则基于环面上嵌套的光滑原子构造，该构造改编自实直线上的对应构造。
英文摘要：
We establish a sharp well-posedness and norm inflation theory for the Camassa--Holm equation in critical Triebel--Lizorkin $F^{1+1/p}_{p,q}(\mathbb{T})$. At the endpoint $p=1$, we prove local Hadamard well-posedness for $1\le q<\infty$. In contrast, we prove norm inflation for $1<p<\infty$ and $1\le q\le\infty$. We also complement the local well-posedness in the critical Besov spaces and higher-regularity Triebel--Lizorkin spaces. The positive results rely on a Lipschitz stability theorem for the periodic Green operator under degree-one Lagrangian flows. The negative result is based on a nested smooth atomic construction on the torus, adapted from its real-line counterpart.
