In this article, we consider a non-local variant of the Kuramoto-Sivashinsky equation in three dimensions (2D interface). Besides showing the global wellposedness of this equation we also obtain some qualitative properties of the solutions. In particular, we prove that the solutions become analytic in the spatial variable for positive time, the existence of a compact global attractor and an upper bound on the number of spatial oscillations of the solutions. We observe that such a bound is particularly interesting due to the chaotic behavior of the solutions.

中文翻译：

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关于3D带电降膜的动力学

在本文中，我们考虑了三维（2D接口）中Kuramoto-Sivashinsky方程的一个非局部变量。除了显示该方程的整体适定性之外，我们还获得了该解决方案的一些定性性质。特别是，我们证明了解决方案可以在正时的空间变量中进行解析，存在紧凑的全局吸引子，并且解决方案的空间振动次数有上限。我们观察到，由于解的混沌行为，这样的界特别有趣。