Boundary-Domain Integral Equations equivalent to an exterior mixed BVP for the variable-viscosity compressible Stokes PDEs

Sergey E. Mikhailov; Carlos F. Portillo

doi:10.3934/cpaa.2021009

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2021, Volume 20, Issue 3: 1103-1133. Doi: 10.3934/cpaa.2021009

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Boundary-Domain Integral Equations equivalent to an exterior mixed BVP for the variable-viscosity compressible Stokes PDEs

Sergey E. Mikhailov^1, , and
Carlos F. Portillo^2,

1.
Department of Mathematics, Brunel University London, UK
2.
Department of Quantitative Methods, Universidad Loyola Andalucía, Sevilla, Spain

^* Corresponding author
^* Corresponding author

Received: July 31, 2020

Revised: October 31, 2020

Early access: January 2021

Published: March 2021

This research was supported by the grants EP/H020497/1, EP/M013545/1, and 1636273 from the EPSRC, UK, and also by Brunel University London

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Abstract

Two direct systems of Boundary-Domain Integral Equations, BDIEs, associated with a mixed boundary value problem for the stationary compressible Stokes system with variable viscosity coefficient in an exterior domain of $ \mathbb{R}^3 $ are derived. This is done by employing the Stokes surface and volume potentials based on an appropriate parametrix (Levi function) in the third Green identities for the velocity and pressure. Mapping properties of the potentials in weighted Sobolev spaces are analysed. Finally, the equivalence between the BDIE systems and the BVP is shown and the isomorphism of operators defined by the BDIE systems is proved.

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Mathematics Subject Classification: Primary: 31B10, 35Q30, 5J57, 45F15; Secondary: 76D07.

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