In this paper we show the local and global well-posedness for the periodic Cauchy problem associated with a special class of 1D Boussinesq systems that emerges in the study of the evolution of long water waves with small amplitude in the presence of surface tension. By a variational approach, we establish the existence of periodic travelling waves. We see that those periodic solutions are characterized as critical points of some functional, for which the existence of critical points follows as a consequence of the ArzelaAscoli Theorem and the fact that the action functional associated is coercive and is (sequentially) weakly lower semi-continuous in an appropriate set.

中文翻译：

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一类一维Boussinesq系统的周期解

在本文中，我们展示了与一类特殊的一维 Boussinesq 系统相关的周期性柯西问题的局部和全局适定性，该系统出现在研究表面张力存在下小振幅长水波的演化过程中。通过变分方法，我们建立了周期性行波的存在。我们看到这些周期解被表征为某些泛函的临界点，对于这些泛函，临界点的存在遵循 ArzelaAscoli 定理以及相关联的动作泛函是强制性的并且（顺序）弱低半连续的事实在适当的集合中。