On the dynamics of 3D electrified falling films

Jiao He; Rafael Granero-Belinchón

doi:10.3934/dcds.2021027

Article Contents

2021, Volume 41, Issue 9: 4041-4064. Doi: 10.3934/dcds.2021027

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On the dynamics of 3D electrified falling films

Jiao He^1, , and
Rafael Granero-Belinchón^2,

1.
Laboratoire de Mathématiques et Modélisation d'Évry, University of Evry & Paris Saclay, Evry, France
2.
Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Santander, Spain

^* Corresponding author: Jiao He
^* Corresponding author: Jiao He

Received: August 2020

Revised: December 2020

Early access: January 2021

Published: September 2021

The first author is supported by the Sophie Germain post-doc program of the Fondation Mathématique Jacques Hadamard

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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.

Keywords:

Kuramoto-Sivashinsky equation,
global wellposedness,
analyticity,
global attractor,
upper bound on the number of spatial oscillations.

Mathematics Subject Classification: Primary: 35K25; 35Q35; Secondary: 37L30.

Citation:

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Figure 1. Initial data (16)

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Figure 2. Numerical solution of (17) for $ \beta = 2, \delta = 0.5, \epsilon = 1 $ at $ t = 40 $

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Figure 3. Numerical solution profile along $ x $-direction for $ y = 0 $ of (17) with $ \beta = 2, \delta = 0.5, \epsilon = 1 $ and the same initial data as in Figure 2 at $ t = 0, 20, 40 $

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Figure 4. Numerical solution profile along $ y $-direction for $ x = \pi/2 $ of (17) with $ \beta = 2, \delta = 0.5, \epsilon = 1 $ and the same initial data as in Figure 2 at $ t = 0, 20, 40 $

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