关于旋转双分量$b$-族浅水波系统：推导与波浪破碎分析
On the rotation-two-component $b$-family shallow-water system: derivation and wave-breaking analysis
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中文总结 AI 辅助
本文从Green--Naghdi型方程推导旋转双分量$b$-族浅水波系统，引入辅助变量建立局部$L^\infty$估计，克服无守恒量困难，给出两个爆破准则，并推广至周期情形。
中文摘要 AI 辅助
本文从Green--Naghdi型方程出发，推导出包含科里奥利力和常涡度效应的旋转双分量$b$-族浅水波系统。该系统在其特殊情形中包含双分量Degasperis--Procesi（2DP）系统作为约化，从而证实了其作为$b$-族旋转扩展的真实角色。随后我们聚焦于一个特定约化，即旋转双分量Degasperis--Procesi系统，其特殊意义在于它结合了两个独立的困难来源：与双分量Camassa-Holm型系统不同，它不具有守恒的$H^1\times L^2$型量；且与标准2DP系统相比，它包含由科里奥利力引起的高阶非线性耦合项。为克服这一双重困难，我们引入一个辅助变量，并在不使用任何守恒律的情况下建立$u$的局部$L^\infty$估计，基于此我们推导出两个不同的爆破准则。当科里奥利力消失时，我们的结果恢复2DP系统的已知准则，并为2DP系统和Degasperis--Procesi方程提供新准则。这些结果自然推广到周期情形及旋转$b$-族系统的相关特殊情形。
英文摘要
In this paper, we derive the rotation-two-component $b$-family shallow-water wave system from the Green--Naghdi type equations incorporating the effects of the Coriolis force and constant vorticity. The system admits, among its special cases, the two-component Degasperis--Procesi (2DP) system as a reduction, thereby confirming its genuine role as a rotational extension of the $b$-family. We then focus on a specific reduction, the rotation-two-component Degasperis-Procesi system, whose particular interest lies in the fact that it combines two independent sources of difficulty: unlike the two-component Camassa-Holm type system, it possesses no conserved $H^1\times L^2$-type quantity, and in contrast to the standard 2DP system, it contains a higher-order nonlinear coupling term induced by the Coriolis force. To overcome this double difficulty, we introduce an auxiliary variable and establish a local $L^\infty$-estimate for $u$ without using any conservation law, based on which we derive two distinct blow-up criteria. When the Coriolis force vanishes, our results recover known criteria for the 2DP system and also provide new criteria for both the 2DP system and Degasperis-Procesi equation. The results extend naturally to the periodic setting and to relevant special cases of the rotational $b$-family system.
发表机构
- School of Mathematics, China University of Mining and Technology（中国矿业大学数学学院）
- Jiangsu Center for Applied Mathematics (CUMT)（江苏应用数学中心（中国矿业大学））
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