On the Boundary-Domain Integral Equation Method for the Dirichlet Problem of the Compressible Stokes System with Variable Viscosity

TL;DR

This paper develops boundary-domain integral equations for the Dirichlet problem of the compressible Stokes system with variable viscosity, providing a new analytical framework for solving such fluid dynamics problems.

Contribution

It introduces two new systems of boundary-domain integral equations using an explicit parametrix, and proves their well-posedness considering the kernels of the involved potentials.

Findings

Established existence and uniqueness of solutions for the BDIE systems.

Analyzed the mapping properties of hydrodynamic potential operators.

Provided a theoretical foundation for numerical methods in variable viscosity Stokes flows.

Abstract

We derive two systems of boundary-domain integral equations (BDIEs) equivalent to the Dirichlet problem for the compressible Stokes system using the potential method with an explicit parametrix (Levi function). The BDIEs are given in terms of surface and volume hydrodynamic potentials. We study the mapping properties of these integral potential operators and used them to prove existence and uniqueness of solution of the two systems of BDIEs obtained taking into account the non-trivial kernels of the single layer and hypersingular hydrodynamic surface potentials.