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November  2019, 18(6): 3059-3088. doi: 10.3934/cpaa.2019137

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

 1 School of Engineering, Computing and Mathematics, Wheatley Campus, Oxford Brookes University, OX33 1HX, Wheatley, UK 2 Department of Mathematics, Brunel University London, UB8 3PH, Uxbridge, UK

* Corresponding author

Received  October 2018 Revised  January 2019 Published  May 2019

Fund Project: This research was supported by the grants EP/H020497/1, EP/M013545/1, and 1636273 from the EPSRC

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.

Citation: Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137
References:
 [1] O. Chkadua, S. E. Mikhailov and D. Natroshvili, Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, Ⅰ: Equivalence and invertibility, J. Integral Equations Appl., 21 (2009), 499-543. doi: 10.1216/JIE-2009-21-4-499. [2] O. Chkadua, S. E. Mikhailov and D. Natroshvili, Analysis of some localized boundary-domain integral equations, J. Integral Equations Appl., 21 (2009), 405-445. doi: 10.1216/JIE-2009-21-3-407. [3] M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal., 19 (1988), 613-626. doi: 10.1137/0519043. [4] G. Eskin, Boundary Value Problems for Elliptic Pseudodifferential Equations, Transl. of Mathem. Monographs, Amer. Math. Soc., vol. 52: Providence, Rhode Island, 1981. [5] R. Gutt, M. Kohr, S. E. Mikhailov and W. L. Wendland, On the mixed problem for the semilinear Darcy-Forchheimer-Brinkman PDE system in Besov spaces on creased Lipschitz domains, Math. Methods in Appl. Sci., 40 (2017), 7780-7829. doi: 10.1002/mma.4562. [6] R. Grzhibovskis, S. Mikhailov and S. Rjasanow, Numerics of boundary-domain integral and integro-differential equations for BVP with variable coefficient in 3D, Computational Mechanics, 51 (2013), 495-503. doi: 10.1007/s00466-012-0777-8. [7] D. Hilbert, Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen, Teubner, Leipzig-Berlin, 2nd edition, 1924. [8] G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, Springer, Berlin, 2008. doi: 10.1007/978-3-540-68545-6. [9] M. Kohr and W. L. Wendland, Variational boundary integral equations for the Stokes system, Applicable Anal., 85 (2006), 1343-1372. doi: 10.1080/00036810600963961. [10] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon & Breach, New York, 1969. [11] E. E. Levi, Ⅰ problemi dei valori al contorno per le equazioni lineari totalmente ellittiche alle derivate parziali, Mem. Soc. Ital. dei Sc. XL, 16 (1909), 1-112. [12] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer, Berlin, 1973. [13] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000. [14] S. G. Michlin and S. Prössdorf, Singular Integral Operators, Springer Berlin, 1986. doi: 10.1007/978-3-642-61631-0. [15] S. E. Mikhailov, Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains, J. Math. Anal. Appl., 378 (2011), 324-342. doi: 10.1016/j.jmaa.2010.12.027. [16] S. E. Mikhailov, Localized boundary-domain integral formulations for problems with variable coefficients, Engineering Analysis with Boundary Elements, 26 (2002), 681-690. [17] S. E. Mikhailov and N. A. Mohamed, Numerical solution and spectrum of boundary-domain integral equation for the Neumann BVP with variable coefficient, Internat. J. Comput. Math., 89 (2012), 1488-1503. doi: 10.1080/00207160.2012.679733. [18] S. E. Mikhailov and I. S. Nakhova, Mesh-based numerical implementation of the localized boundary-domain integral equation method to a variable-coefficient Neumann problem, J. Eng. Math., 51 (2005), 251-259. doi: 10.1007/s10665-004-6452-0. [19] S. E. Mikhailov and C. F. Portillo, BDIE system to the mixed BVP for the Stokes equations with variable viscosity, In Integral Methods in Science and Engineering: Theoretical and Computational Advances, C. Constanda and A. Kirsh, eds., Springer, Boston, (2015), 401–412. [20] S. E. Mikhailov and C. F. Portillo, Analysis of boundary-domain integral equations based on a new paramatrix for the mixed diffusion BVP with variable coefficient in an interior Lipschitz domain, Journal of Integral Equations and Applications, forthcoming (2018). Available at https://projecteuclid.org:443/euclid.jiea/1541668069. [21] C. Miranda, Partial Differential Equations of Elliptic Type, 2nd edn., Springer, 1970. [22] A. Pomp, Levi functions for linear elliptic systems with variable coefficients including shell equations, Comput. Mech., 22 (1998), 93-99. doi: 10.1007/s004660050343. [23] A. Pomp, The Boundary-domain Integral Method for Elliptic Systems. With Applications in Shells, volume 1683 of Lecture Notes in Mathematics., Springer, Berlin-Heidelberg-New York, 1998. doi: 10.1007/BFb0094576. [24] B. Reidinger and O. Steinbach, A symmetric boundary element method for the Stokes problem in multiple connected domains, Math. Meth. Appl. Sci., 26 (2003), 77-93. doi: 10.1002/mma.347. [25] C. Le Roux and B. D. Reddy, The steady Navier-Stokes equations with mixed boundary conditions: application to free boundary flows, Nonlinear Analysis, Theory, Methods & Applications, 20 (1993), 1043-1068. doi: 10.1016/0362-546X(93)90094-9. [26] J. Sladek, V. Sladek and S. N. Atluri, Local boundary integral equation (LBIE) method for solving problems of elasticity with nonhomogeneous material properties, Comput. Mech., 24 (2000), 456-462. [27] J. Sladek, V. Sladek and J.-D. Zhang, 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. doi: 10.1007/s10665-004-3692-y. [28] O. Steinbach, Numerical Approximation Methods for Elliptic Boundary Value Problems, Springer Berlin, 2007. doi: 10.1007/978-0-387-68805-3. [29] A. E. Taigbenu, The Green Element Method, Kluwer Academic Publishers, Boston-Dordrecht-London, 1999. [30] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978. [31] W. L. Wenland and J. Zhu, The boundary element method for three dimensional Stokes flow exterior to an open surface, Mathematical and Computer Modelling, 6 (1991), 19-42. doi: 10.1016/0895-7177(91)90021-X. [32] T. Zhu, J.-D. Zhang and S. N. Atluri, A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach, Comput. Mech., 21 (1998), 223-235. doi: 10.1007/s004660050297. [33] T. Zhu, J.-D. Zhang and S. N. Atluri, A meshless numerical method based on the local boundary integral equation (LBIE) to solve linear and non-linear boundary value problems, Eng. Anal. Bound. Elem., 23 (1999), 375-389.

show all references

References:
 [1] O. Chkadua, S. E. Mikhailov and D. Natroshvili, Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, Ⅰ: Equivalence and invertibility, J. Integral Equations Appl., 21 (2009), 499-543. doi: 10.1216/JIE-2009-21-4-499. [2] O. Chkadua, S. E. Mikhailov and D. Natroshvili, Analysis of some localized boundary-domain integral equations, J. Integral Equations Appl., 21 (2009), 405-445. doi: 10.1216/JIE-2009-21-3-407. [3] M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal., 19 (1988), 613-626. doi: 10.1137/0519043. [4] G. Eskin, Boundary Value Problems for Elliptic Pseudodifferential Equations, Transl. of Mathem. Monographs, Amer. Math. Soc., vol. 52: Providence, Rhode Island, 1981. [5] R. Gutt, M. Kohr, S. E. Mikhailov and W. L. Wendland, On the mixed problem for the semilinear Darcy-Forchheimer-Brinkman PDE system in Besov spaces on creased Lipschitz domains, Math. Methods in Appl. Sci., 40 (2017), 7780-7829. doi: 10.1002/mma.4562. [6] R. Grzhibovskis, S. Mikhailov and S. Rjasanow, Numerics of boundary-domain integral and integro-differential equations for BVP with variable coefficient in 3D, Computational Mechanics, 51 (2013), 495-503. doi: 10.1007/s00466-012-0777-8. [7] D. Hilbert, Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen, Teubner, Leipzig-Berlin, 2nd edition, 1924. [8] G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, Springer, Berlin, 2008. doi: 10.1007/978-3-540-68545-6. [9] M. Kohr and W. L. Wendland, Variational boundary integral equations for the Stokes system, Applicable Anal., 85 (2006), 1343-1372. doi: 10.1080/00036810600963961. [10] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon & Breach, New York, 1969. [11] E. E. Levi, Ⅰ problemi dei valori al contorno per le equazioni lineari totalmente ellittiche alle derivate parziali, Mem. Soc. Ital. dei Sc. XL, 16 (1909), 1-112. [12] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer, Berlin, 1973. [13] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000. [14] S. G. Michlin and S. Prössdorf, Singular Integral Operators, Springer Berlin, 1986. doi: 10.1007/978-3-642-61631-0. [15] S. E. Mikhailov, Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains, J. Math. Anal. Appl., 378 (2011), 324-342. doi: 10.1016/j.jmaa.2010.12.027. [16] S. E. Mikhailov, Localized boundary-domain integral formulations for problems with variable coefficients, Engineering Analysis with Boundary Elements, 26 (2002), 681-690. [17] S. E. Mikhailov and N. A. Mohamed, Numerical solution and spectrum of boundary-domain integral equation for the Neumann BVP with variable coefficient, Internat. J. Comput. Math., 89 (2012), 1488-1503. doi: 10.1080/00207160.2012.679733. [18] S. E. Mikhailov and I. S. Nakhova, Mesh-based numerical implementation of the localized boundary-domain integral equation method to a variable-coefficient Neumann problem, J. Eng. Math., 51 (2005), 251-259. doi: 10.1007/s10665-004-6452-0. [19] S. E. Mikhailov and C. F. Portillo, BDIE system to the mixed BVP for the Stokes equations with variable viscosity, In Integral Methods in Science and Engineering: Theoretical and Computational Advances, C. Constanda and A. Kirsh, eds., Springer, Boston, (2015), 401–412. [20] S. E. Mikhailov and C. F. Portillo, Analysis of boundary-domain integral equations based on a new paramatrix for the mixed diffusion BVP with variable coefficient in an interior Lipschitz domain, Journal of Integral Equations and Applications, forthcoming (2018). Available at https://projecteuclid.org:443/euclid.jiea/1541668069. [21] C. Miranda, Partial Differential Equations of Elliptic Type, 2nd edn., Springer, 1970. [22] A. Pomp, Levi functions for linear elliptic systems with variable coefficients including shell equations, Comput. Mech., 22 (1998), 93-99. doi: 10.1007/s004660050343. [23] A. Pomp, The Boundary-domain Integral Method for Elliptic Systems. With Applications in Shells, volume 1683 of Lecture Notes in Mathematics., Springer, Berlin-Heidelberg-New York, 1998. doi: 10.1007/BFb0094576. [24] B. Reidinger and O. Steinbach, A symmetric boundary element method for the Stokes problem in multiple connected domains, Math. Meth. Appl. Sci., 26 (2003), 77-93. doi: 10.1002/mma.347. [25] C. Le Roux and B. D. Reddy, The steady Navier-Stokes equations with mixed boundary conditions: application to free boundary flows, Nonlinear Analysis, Theory, Methods & Applications, 20 (1993), 1043-1068. doi: 10.1016/0362-546X(93)90094-9. [26] J. Sladek, V. Sladek and S. N. Atluri, Local boundary integral equation (LBIE) method for solving problems of elasticity with nonhomogeneous material properties, Comput. Mech., 24 (2000), 456-462. [27] J. Sladek, V. Sladek and J.-D. Zhang, 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. doi: 10.1007/s10665-004-3692-y. [28] O. Steinbach, Numerical Approximation Methods for Elliptic Boundary Value Problems, Springer Berlin, 2007. doi: 10.1007/978-0-387-68805-3. [29] A. E. Taigbenu, The Green Element Method, Kluwer Academic Publishers, Boston-Dordrecht-London, 1999. [30] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978. [31] W. L. Wenland and J. Zhu, The boundary element method for three dimensional Stokes flow exterior to an open surface, Mathematical and Computer Modelling, 6 (1991), 19-42. doi: 10.1016/0895-7177(91)90021-X. [32] T. Zhu, J.-D. Zhang and S. N. Atluri, A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach, Comput. Mech., 21 (1998), 223-235. doi: 10.1007/s004660050297. [33] T. Zhu, J.-D. Zhang and S. N. Atluri, A meshless numerical method based on the local boundary integral equation (LBIE) to solve linear and non-linear boundary value problems, Eng. Anal. Bound. Elem., 23 (1999), 375-389.
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