# On the dynamics of 3D electrified falling films

## TL;DR

This paper studies a 3D non-local Kuramoto-Sivashinsky equation, proving well-posedness, analyticity, existence of a global attractor, and bounds on oscillations, shedding light on the chaotic dynamics of electrified falling films.

## Contribution

It introduces a non-local 3D model for electrified falling films and establishes fundamental mathematical properties including well-posedness and long-term behavior.

## Key findings

- Solutions become analytic in space for positive time
- Existence of a compact global attractor
- Bound on the number of spatial oscillations

## Abstract

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.