Skip to main content
SpringerLink
Log in

On the Well-Posedness of Two Boundary-Domain Integral Equation Systems Equivalent to the Dirichlet Problem for the Stokes System with Variable Viscosity

  • Published:
Results in Mathematics Aims and scope Submit manuscript

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 the surface and volume hydrodynamic potentials. The mapping properties of these integral potential operators are analysed and applied 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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Availability of data and material (data transparency)

Not applicable.

References

  1. Ayele, T.G., Dagnaw, M.A.: Boundary-domain integral equation systems to the Dirichlet and Neumann problems for compressible Stokes equations with variable viscosity in 2D. Math. Meth. Appl. Sci. (2020), 1-23

  2. Beshley, A., Chapko, B., Johansson, T.: On the alternating method and boundary-domain integrals for elliptic Cauchy problems. Comput. Math. Appl. (2019). https://doi.org/10.1016/j.camwa.2019.05.025

    Article  MathSciNet  MATH  Google Scholar 

  3. Beshley, A., Chapko, R., Johansson, B.T.: An integral equation method for the numerical solution of a Dirichlet problem for second-order elliptic equations with variable coefficients. J. Eng. Math. 112, 63–73 (2018)

    Article  MathSciNet  Google Scholar 

  4. Chkadua, O., Mikhailov, S.E., Natroshvili, D.: Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, I: Equivalence and invertibility. J. Integral Equ. Appl. 21, 499–543 (2009)

    MathSciNet  MATH  Google Scholar 

  5. Choi, J., Dong, H., Kim, D.: Green functions of conormal derivative problems for stationary Stokes system. J. Math. Fluid Mech. 20, 1745–1769 (2018). https://doi.org/10.1007/s00021-018-0387-0

    Article  MathSciNet  MATH  Google Scholar 

  6. Costabel, M.: Boundary integral operators on Lipschitz domains: Elementary results. SIAM J. Math. Anal. 19, 613–626 (1988)

    Article  MathSciNet  Google Scholar 

  7. Fikl, A., Bodony, D.J.: Jump relations of certain hypersingular Stokes kernels on regular surfaces. SIAM J. Appl. Math. (2020). https://doi.org/10.1137/19M1269804

    Article  MathSciNet  MATH  Google Scholar 

  8. Fresneda-Portillo, C., Woldemicheal, Z.W.: A New Parametrix for the boundary-domain integral equations for the Dirichlet problem of the diffusion equation in inhomogeneous media with \(H^{-1}. (\Omega )\) right-hand side on Lipschitz Domains. Math. Methods. Appl. Sci. (2020). https://doi.org/10.1002/mma.6659

    Article  MATH  Google Scholar 

  9. Fresneda-Portillo, C., Woldemicheal, Z.W.: On the existence of solution of the boundary-domain integral equation system derived from the 2D Dirichlet problem for the diffusion equation with variable coefficient using a novel parametrix. Complex Var. Elliptic Equ. (2019). https://doi.org/10.1080/17476933.2019.1687457

    Article  MATH  Google Scholar 

  10. Fresneda-Portillo, C.: Boundary-Domain Integral Equations for the diffusion equation in inhomogeneous media based on a new family of parametrices. Complex Var. Elliptic Eq. (2019). https://doi.org/10.1080/17476933.2019.1591382

    Article  MathSciNet  MATH  Google Scholar 

  11. Fresneda-Portillo, C., Mikhailov, S.E.: Analysis of boundary-domain integral equations to the mixed BVP for a compressible Stokes system with variable viscosity. Commun. Pure Appl. Anal. 18, 3059–3088 (2019)

    Article  MathSciNet  Google Scholar 

  12. Grzhibovskis, R., Mikhailov, S.E., Rjasanow, S.: Numerics of boundary-domain integral and integro-differential equations for BVP with variable coefficient in 3D. Comput. Mech. 51, 495–503 (2013)

    Article  MathSciNet  Google Scholar 

  13. Hsiao, G.C., Wendland, W.L.: Boundary Integral Equations. Springer, Berlin (2008)

    Book  Google Scholar 

  14. Kohr, M., Wendland, W.L.: Variational boundary integral equations for the Stokes system. Appl. Anal. 85, 1343–1372 (2006)

    Article  MathSciNet  Google Scholar 

  15. Kohr, M.: Boundary value problems for a compressible Stokes system in bounded domains in Rn. J. Comput. Appl. Math. 201, 128–145 (2007). https://doi.org/10.1016/j.cam.2006.02.004

    Article  MathSciNet  MATH  Google Scholar 

  16. Kohr, M., Mikhailov, S.E., Wendland, W.L.: Potentials and transmission problems in weighted Sobolev spaces for anisotropic Stokes and Navier-Stokes systems with \(L^{\infty }\) strongly elliptic coefficient tensor. Complex Var. Elliptic Equ. 65(1), 109–140 (2020). https://doi.org/10.1080/17476933.2019.1631293

    Article  MathSciNet  MATH  Google Scholar 

  17. Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon & Breach, New York (1969)

    MATH  Google Scholar 

  18. Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications. Springer, Berlin (1973)

    Book  Google Scholar 

  19. McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  20. Mikhailov, S.E.: Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains. J. Math. Anal. Appl. 378, 324–342 (2011)

    Article  MathSciNet  Google Scholar 

  21. Pomp, A.: The Boundary-Domain Integral Method for Elliptic Systems: With Application to Shells. Springer, Berlin (1998)

    Book  Google Scholar 

  22. Ravnik, J., Tibaut, J.: Fast boundary-domain integral method for unsteady convection-diffusion equation with variable diffusivity using the modified Helmholtz fundamental solution. Numer. Algor. 82, 1441–1466 (2019). https://doi.org/10.1007/s11075-019-00664-3

    Article  MathSciNet  MATH  Google Scholar 

  23. Reidinger, B., Steinbach, O.: A symmetric boundary element method for the Stokes problem in multiple connected domains. Math. Meth. Appl. Sci. 26, 77–93 (2003)

    Article  MathSciNet  Google Scholar 

  24. Sladek, J., Sladek, V., Zhang, Ch.: Local integro-differential equations with domain elements for the numerical solution of partial differential equations with variable coefficients. J. Eng. Math. 51(2005), 261–282 (2005)

    Article  MathSciNet  Google Scholar 

  25. Steinbach, O.: Numerical Approximation Methods for Elliptic Boundary Value Problems. Springer, Berlin (2007)

    Google Scholar 

  26. Tibaut, J., Ravnik, J.: Fast boundary-domain integral method for heat transfer simulations. Eng. Anal. Bound. Elem. 99, 222–232 (2019). https://doi.org/10.1016/j.enganabound.2018.12.003

    Article  MathSciNet  MATH  Google Scholar 

  27. Walter, J., Salsac, A.-V., Barthès Biesel, D., Le Tallec, P.: Coupling of finite element and boundary integral methods for a capsule in a Stokes flow. Int. J. Numer. Meth. Engng. 83, 829–850 (2010). https://doi.org/10.1002/nme.2859

    Article  MathSciNet  MATH  Google Scholar 

  28. Wenland, W.L., Zhu, J.: The boundary element method for three dimensional Stokes flow exterior to an open surface. Math. Comput. Modell. 6, 19–42 (1991)

    Article  MathSciNet  Google Scholar 

Download references

Funding

Not applicable.

Author information

Authors and Affiliations

  1. Department of Mathematics, Injibara University, Injibara, Ethiopia

    M. A. Dagnaw

  2. Department of Quantitative Methods, Universidad Loyola Andalucía, Campus Sevilla, 41404, Dos Hermanas, Sevilla, Spain

    C. Fresneda-Portillo

Authors
  1. M. A. Dagnaw
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Fresneda-Portillo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. Fresneda-Portillo.

Ethics declarations

Conflicts of interest

Not applicable.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dagnaw, M.A., Fresneda-Portillo, C. On the Well-Posedness of Two Boundary-Domain Integral Equation Systems Equivalent to the Dirichlet Problem for the Stokes System with Variable Viscosity. Results Math 77, 99 (2022). https://doi.org/10.1007/s00025-022-01630-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-022-01630-7

Mathematics Subject Classification

Search

Navigation