On quaternionic functions for the solution of an ill-posed Cauchy problem for a viscous fluid

Abstract

Holomorphic functions are the key tool to construct representation formulae for the solutions for a manifold of plane problems, especially for the flow of a viscous fluid modelled by the Stokes system. Three-dimensional representation formulae can be constructed by tools of hypercomplex analysis, i.e. by working with monogenic functions playing the role of a three-dimensional analogue of holomorphic functions. However, several alternative constructions in hypercomplex setting are possible. In this paper, the three-dimensional representation of a general solution for the Stokes system, based on the functions of a reduced quaternionic variable, is presented. Moreover, an ill-posed Cauchy problem for the Stokes system, consisting in reconstruction of the velocity field in the interior from overdetermined boundary conditions given on a part of the boundary, is considered. It is shown, that if the domain is star-shaped, then the Cauchy problem can be reduced to the problem of the regular extension for a quaternionic function from the boundary conditions given on a part of its boundary.Holomorphic functions are the key tool to construct representation formulae for the solutions for a manifold of plane problems, especially for the flow of a viscous fluid modelled by the Stokes system. Three-dimensional representation formulae can be constructed by tools of hypercomplex analysis, i.e. by working with monogenic functions playing the role of a three-dimensional analogue of holomorphic functions. However, several alternative constructions in hypercomplex setting are possible. In this paper, the three-dimensional representation of a general solution for the Stokes system, based on the functions of a reduced quaternionic variable, is presented. Moreover, an ill-posed Cauchy problem for the Stokes system, consisting in reconstruction of the velocity field in the interior from overdetermined boundary conditions given on a part of the boundary, is considered. It is shown, that if the domain is star-shaped, then the Cauchy problem can be reduced to the problem of the regular extension for a quater...