The aim of this paper is to study the influence of the vorticity on the existence time in fluid systems for which global smoothness and decay are known in the case of small irrotational data. We focus on two examples: the Euler–Korteweg system and the two-fluid Euler–Maxwell system. We prove that the lower bound of the lifespan of these systems is no less than the inverse of the $H^s$ $(s>5∕2)$ norm of the rotational part of the initial velocity. Our approach is based on energy estimates and the fast time decay results of global solutions to these systems with irrotational initial data.

中文翻译：

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带有涡度的Euler-Korteweg方程和两流体Euler-Maxwell方程的大时间存在

本文的目的是研究涡旋对流体系统中存在时间的影响，在小旋转数据的情况下，流体系统的整体光滑度和衰减是已知的。我们关注两个例子：欧拉-柯特维格系统和双流体欧拉-麦克斯韦系统。我们证明这些系统寿命的下限不小于该系统寿命的倒数。$H^s$ $（s>5∕2）$初始速度的旋转部分的范数。我们的方法是基于能量估计和具有无旋转初始数据的这些系统的整体解决方案的快速时间衰减结果。