Euler equation in a channel in space dimension 2 and 3

Pages: 755 - 778, Volume 13, Issue 3, August 2005

doi:10.3934/dcds.2005.13.755       Abstract        Full Text (323.0K)       Related Articles

M. Petcu - Laboratoire d'Analyse Numérique, Université de Paris-Sud, Orsay, The Institute of Mathematics of the Romanian Academy, Bucharest, Romania, The Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN, United States (email)

Abstract: In this article we consider the Euler equations of an ideal incompressible fluid in a $2D$ and $3D$ channel and we prove the existence and uniqueness of classical solutions for all time for the $2D$ case and the local in time existence for the $3D$ case. For the $2D$ case, the proof makes use of the Schauder fixed point, and specific properties of the Green function in a channel are derived. For the $3D$ case, we use a priori estimates on some appropriate Sobolev spaces and the existence of solution follows by the Galerkin method.

Keywords:  Euler equation, energy estimate, Green function, Schauder fixed point theorem.
Mathematics Subject Classification:  35Q35, 76B03.

Received: September 2004;      Revised: March 2005;      Published: May 2005.