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Solvability of an initial boundary value problem for the euler equations in twodimensional domain with corners
Author(s) -
Zajaczkowski W. M.,
Piskorek A.
Publication year - 1984
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670060102
Subject(s) - mathematics , uniqueness , mathematical analysis , sobolev space , boundary value problem , domain (mathematical analysis) , bounded function , section (typography) , neumann boundary condition , smoothness , euler equations , dirichlet problem , boundary (topology) , dirichlet distribution , advertising , business
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 ⊂ R 2 with corners. We consider two cases of a nonvanishing normal component of velocity on the boundary. In three‐dimensional 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.