Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity

Abstract

The mixed boundary value problem for a compressible Stokes system of partial differential equations in a bounded domain is reduced to two different systems of segregated direct Boundary-Domain Integral Equations (BDIEs) expressed in terms of surface and volume parametrix-based potential type operators. Equivalence of the BDIE systems to the mixed BVP and invertibility of the matrix operators associated with the BDIE systems are proved in appropriate Sobolev spaces.2000 Mathematics Subject Classification. Primary: 35J57, 45F15; Secondary: 45P05.

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

“…Choosing the right parametrix is crucial in order to establish relatively simple relationships of the surface and volume potentials with their counterparts in the constant coefficient case, which is essential in proving the equivalence and invertibility theorems. The boundary-domain integral equations to the mixed BVP in bounded domains for the compressible Stokes system with variable viscosity have been investigated in [9] (see also [25,27] for the incompressible case).…”

“…In this paper, we derive two direct BDIE systems associated with the considered mixed boundary value problem for the stationary compressible Stokes system with variable viscosity, defined in an exterior domain of R 3 . This is done by employing the Stokes surface and volume potentials based on the parametrix (Levi function) used in [25,27,9] in the third Green identities for the velocity and pressure. Then we analyse mapping properties of the potentials in weighted Sobolev spaces.…”

Dirichlet and transmission problems for anisotropic stokes and Navier-Stokes systems with <i>L</i>_∞ tensor coefficient under relaxed ellipticity condition

“…An alternative approach, which reduces various boundary problems for variable coefficient elliptic partial differential equations to boundary-domain integral equations (BDIEs), by means of explicit parametrix-based integral potentials, was explored, e.g., in [13,14,40] and the references therein.…”

A new family of boundary-domain integral equations for the diffusion equation with variable coefficient in unbounded domains

“…To overcome this obstacle, one can construct a parametrix using the known fundamental solution. A discussion on fundamental solution existence theorems, algorithms for constructing fundamental solutions and parametrices is available in [24]; for classical examples of derivation of Boundary Domain Integral Equations refer to [7] for the diffusion equation with variable coefficient in bounded domains in R 3 ; [25] for the same problem applying a different parametrix; [26] for the Dirichlet problem in R 2 and [22] for the mixed problem for the compressible Stokes system, as an example of derivation of BDIEs from a PDE system.…”

“…However, for some PDE problems, it is not always possible to obtain a parametrix that depends exclusively on a(y) and not on a(x). This is the case of the Stokes system, see [22]. Hence, the usefulness of the analysis of the family of parametrices depending on a(x).…”