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The fluid mecanic is a strange world. The equation are in general gouverned by parameters, which determined the solution. Imagine that the solution is known for precise values of those parameters, and correspond to experimental results. Now you want the solution for values which are a bit different. The intuition say that the solution is certainly a bit differnt but stay similar to the first one...

The intuition is false, and fluid mecanic prouves that. The theory of instability shows that the solutions can be very different.

In this project, I have studied the Kuramoto-Sivashinski equation which an illustration of the instabilities of the fluid mecanic.

The Karamoto-Sivashinsky equation (KSE) is a one-dimensional partial derivative equation. It's most usual form is :

$\displaystyle \frac{ \partial u}{\partial t}+ u \frac{ \partial u}{\partial x}+...
... \partial^2 u}{{\partial x}^2}+ \lambda \frac{ \partial^4 u}{{\partial x}^4}=0

Firstly, I have theoriticaly studied the stability of this equation, then as the theoretical study become to hard, I have begin the numerical simulation of the KSE. The final part of this report contain the main result I have obtained.

Julien Delbove