发表机构
University of Rochester（罗切斯特大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究超弹性杆方程在临界Sobolev空间$H^{3/2}$中的不适定性，证明端点处不适定，且性质依赖于参数$\gamma$：$0<\gamma\leq3$时小初值有限时间爆破，$\gamma<0$或$\gamma>3$时$H^{3/2}$范数任意短时间内任意大。
AI 中文摘要
本文是研究Camassa-Holm型方程在临界Sobolev空间中不适定性系列文章的第三篇。继$b$-Novikov和Fokas-Olver-Rosenau-Qiao方程的结果之后，我们研究了临界Sobolev正则性$H^{3/2}(\mathbb R)$下的超弹性杆方程。我们证明了在该端点处的不适定性，补充了$s>3/2$时$H^s(\mathbb R)$中的局部适定性理论以及最近建立的$1<s<3/2$时的范数膨胀结果。临界不适定性的性质取决于参数$\gamma$。对于$0<\gamma\leq3$，我们构造任意小的光滑初始数据，其对应解在有限时间内发生爆破，且最大存在时间随初始数据的大小趋于零。对于$\gamma<0$或$\gamma>3$，我们构造具有任意小$H^{3/2}$初始数据的解，其$H^{3/2}$范数在任意短时间内变得任意大。
英文摘要
This article is the third in a series investigating ill-posedness in critical Sobolev spaces for Camassa-Holm-type equations. Following results for the $b$-Novikov and Fokas-Olver-Rosenau-Qiao equations, we study the hyperelastic rod equation at the critical Sobolev regularity $H^{3/2}(\mathbb R)$. We prove ill-posedness at this endpoint, complementing the local well-posedness theory in $H^s(\mathbb R)$ for $s>3/2$ and the recently established norm-inflation result for $1<s<3/2$. The nature of the critical ill-posedness depends on the parameter $γ$. For $0<γ\leq3$, we construct arbitrarily small smooth initial data whose corresponding solutions develop finite-time blow-up, with the maximal lifespan tending to zero with the size of the initial data. For $γ<0$ or $γ>3$, we construct solutions with arbitrarily small $H^{3/2}$ initial data whose $H^{3/2}$ norm becomes arbitrarily large in arbitrarily short time.
Comments13 pages
