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\begin{document}

\title{Boundary-Domain Integral Equation Systems to the Mixed BVP for
Compressible Stokes Equations with Variable Viscosity in 2D}



\author[1]{Tsegaye Ayele}%
\author[2]{Sergey Mikhailov}%
\affil[1]{Addis Ababa University}%
\affil[2]{Brunel University London}%


\vspace{-1em}



  
  \date{May 27, 2024}


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\begin{abstract}
In this paper, the Boundary-Domain Integral Equations (BDIEs) for the
mixed boundary value problem(BVP) for a compressible Stokes system of
partial differential equation(PDE) with variable coefficient in 2D is
considered . An appropriate parametrix is used to reduce this BVP to the
BDIEs. Although the theory of BDIEs in 3D is well developed, the BDIEs
in 2D need a special consideration due to their different equivalence
properties. As a result, we need to set conditions on the domain or on
the spaces to ensure the invertibility of corresponding parametrix-based
integral layer potentials and hence the unique solvability of BDIEs. The
properties of corresponding potential operators are investigated.
Equivalence of the BDIE systems to the mixed BVP and invertibility of
the matrix operators associated with the BDIE systems in appropriate
Sobolev spaces are proved.%
\end{abstract}%



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