AI 中文总结
研究卡马萨 - 霍尔姆方程在临界特里贝尔 - 利佐金空间的适定性与不适定性，通过拉格朗日坐标变换建立局部适定性，利用光滑原子分解证明强不适定性，为相关方程不适定性研究提供新视角。
AI 中文摘要
本文致力于研究卡马萨 - 霍尔姆（CH）方程柯西问题在临界特里贝尔 - 利佐金空间\(F^{1+\frac{1}{p}}_{p,q}(\mathbb{R})\)（其中\((p,q)\in[1,\infty)\times[1,\infty]\)或\(p = q=\infty\)）中的适定性与不适定性。一方面，通过拉格朗日坐标变换在\(F^2_{1,q}(\mathbb{R})\)（\(1\leq q<\infty\)）中建立了哈达玛意义下的局部适定性。另一方面，利用光滑原子分解证明了在\((p,q)\in(1,\infty)\times[1,\infty]\)或\(p = q=\infty\)时\(F^{1+\frac{1}{p}}_{p,q}(\mathbb{R})\)中范数膨胀意义下的强不适定性，这尤其得出了\(1<p<\infty\)时临界索伯列夫空间\(W^{1+\frac{1}{p},p}(\mathbb{R})\)中CH方程的不适定性，并为\(H^{\frac{3}{2}}(\mathbb{R})\)中CH方程的不适定性提供了新视角。
英文摘要
This paper is devoted to the sharp well-posedness and ill-posedness of the Cauchy problem for the Camassa-Holm (CH) equation in critical Triebel-Lizorkin spaces $F^{1+\frac{1}{p}}_{p,q}(\mathbb{R})$ with $(p,q)\in[1,\infty)\times[1,\infty]$ or $p=q=\infty$. On the one hand, we establish the local well-posedness in the sense of Hadamard in $F^2_{1,q}(\mathbb{R})$ for $1\leq q<\infty$ via Lagrangian coordinate transformation. On the other hand, by means of smooth atomic decomposition, strong ill-posedness is then proved in $F^{1+\frac{1}{p}}_{p,q}(\mathbb{R})$ with $(p,q)\in(1,\infty)\times[1,\infty]$ or $p=q=\infty$ in the sense of norm inflation, which in particular yields the ill-posedness of CH in critical Sobolev spaces $W^{1+\frac{1}{p},p}(\mathbb{R})$ with $1<p<\infty$, and provides a new perspective on the ill-posedness of CH in $H^{\frac{3}{2}}(\mathbb{R})$.
Comments14 pages
