论文标题
中间长波方程的深水和浅水极限
Deep-water and shallow-water limits of the intermediate long wave equation
论文作者
论文摘要
在本文中，我们研究了中间长波方程（ILW）的低规律性收敛问题，相对于深度参数$δ> 0 $，在实际线和圆上。作为Korteweg-de Vries（KDV）和Benjamin-Ono（BO）方程之间的天然桥梁，ILW方程非常具有物理意义。我们证明，ILW的解决方案以$ s> \ frac12 $的$ h^s-sobolev空间汇聚给了Bo的深水限制（如$δ\ to \ infty $），以及在浅水限制中的KDV（$δ\ 0 $）。这改善了Abdelouhab，Bona，Felland和Saut（1989）的先前收敛结果，该结果需要在深水限制中$ s> \ frac32 $，在浅水限制中需要$ s \ geq2 $。此外，收敛结果也适用于广义ILW方程，即〜使用$ k \ geq 2 $的非线性$ \ partial_x（u^k）$。此外，这项工作给出了定期性$ s \ geq \ frac34 $的圆圈中广义ILW解决方案的第一个收敛结果。总体而言，这项研究为ILW方程及其解决方案在不同水深中的行为提供了数学见解，并且对预测和建模各种环境中的波浪行为具有影响。
In this paper, we study the low regularity convergence problem for the intermediate long wave equation (ILW), with respect to the depth parameter $δ>0$, on the real line and the circle. As a natural bridge between the Korteweg-de Vries (KdV) and the Benjamin-Ono (BO) equations, the ILW equation is of physical interest. We prove that the solutions of ILW converge in the $H^s$-Sobolev space for $s>\frac12$, to those of BO in the deep-water limit (as $δ\to\infty$), and to those of KdV in the shallow-water limit (as $δ\to 0$). This improves previous convergence results by Abdelouhab, Bona, Felland, and Saut (1989), which required $s>\frac32$ in the deep-water limit and $s\geq2$ in the shallow-water limit. Moreover, the convergence results also apply to the generalised ILW equation, i.e.~with nonlinearity $\partial_x (u^k)$ for $k\geq 2$. Furthermore, this work gives the first convergence results of generalised ILW solutions on the circle with regularity $s\geq \frac34$. Overall, this study provides mathematical insights for the behaviour of the ILW equation and its solutions in different water depths, and has implications for predicting and modelling wave behaviour in various environments.
