Solvability of an initial boundary value problem for the Euler equations in twodimensional domain with corners
Abstract
The aim of this paper is to prove the existence and uniqueness of local solutions of some initial boundary value problems for the Euler equations of an incompressible fluid in a bounded domain Ω ⊂ R2 with corners. We consider two cases of a nonvanishing normal component of velocity on the boundary. In threedimensional case such problems have been considered in papers [12], [13], [14]. Similar problems in domains without corners have been considered in [2][6], [11]. In this paper the relation between the maximal corner angle of the boundary and the smoothness of the solutions is shown. The paper consists of four sections. In section 1 two initial boundary value problems for the Euler equations are formulated. In section 2 the existence and uniqueness of solutions of the Laplace equation in twodimensional domain with corners for the Dirichlet and Neumann problems is proved in the Sobolev spaces. In sections 3 and 4 we prove the existence and uniqueness of solutions of problems formulated in section 1, using the method of successive approximations.
 Publication:

Mathematical Methods in the Applied Sciences
 Pub Date:
 1984
 DOI:
 10.1002/mma.1670060102
 Bibcode:
 1984MMAS....6....1Z
 Keywords:

 Boundary Value Problems;
 Computational Fluid Dynamics;
 Corner Flow;
 Euler Equations Of Motion;
 Two Dimensional Flow;
 Dirichlet Problem;
 Incompressible Fluids;
 Laplace Equation;
 Neumann Problem;
 Fluid Mechanics and Heat Transfer