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April  2005, 13(3): 755-778. doi: 10.3934/dcds.2005.13.755

Euler equation in a channel in space dimension 2 and 3

M. Petcu 1,

1. 

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

Received  September 2004 Revised  March 2005 Published  May 2005

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: energy estimate, Schauder fixed point theorem., Green function, Euler equation.
Mathematics Subject Classification: 35Q35, 76B0.
Citation: M. Petcu. Euler equation in a channel in space dimension 2 and 3. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 755-778. doi: 10.3934/dcds.2005.13.755
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