The Stokes-Bitsadze Problem as a Double Div-Curl Problem

Abstract

The stream function, Airy-stress function formulation of the classical Stokes problem in the plane has not been exhaustively studied as an elliptic boundary value problem. Bitsadze appears to have been the first to question the well-posedness of this formulation subject to certain types of boundary conditions, although it is not known whether Bitsadze was aware that canonical system of equations, named after him (the Bitsadze equations) were mathematically equivalent to the incompressible Stokes equations. In this paper we identify the Stokes-Bitsadze problem as a double div-curl problem. Using results about the Cauchy-Riemann equations we are able to present several existence and uniqueness results for solutions of the Stokes-Bitsadze problem subject to different boundary conditions.