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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. ChkaduaS. 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.  Google Scholar

[2]

O. ChkaduaS. 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.  Google Scholar

[3]

M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal., 19 (1988), 613-626.  doi: 10.1137/0519043.  Google Scholar

[4]

G. Eskin, Boundary Value Problems for Elliptic Pseudodifferential Equations, Transl. of Mathem. Monographs, Amer. Math. Soc., vol. 52: Providence, Rhode Island, 1981.  Google Scholar

[5]

R. GuttM. KohrS. 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.  Google Scholar

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R. GrzhibovskisS. 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.  Google Scholar

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D. Hilbert, Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen, Teubner, Leipzig-Berlin, 2nd edition, 1924.  Google Scholar

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G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, Springer, Berlin, 2008. doi: 10.1007/978-3-540-68545-6.  Google Scholar

[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.  Google Scholar

[10]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon & Breach, New York, 1969.  Google Scholar

[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.   Google Scholar

[12]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer, Berlin, 1973.  Google Scholar

[13]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000.  Google Scholar

[14]

S. G. Michlin and S. Prössdorf, Singular Integral Operators, Springer Berlin, 1986. doi: 10.1007/978-3-642-61631-0.  Google Scholar

[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.  Google Scholar

[16]

S. E. Mikhailov, Localized boundary-domain integral formulations for problems with variable coefficients, Engineering Analysis with Boundary Elements, 26 (2002), 681-690.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[21]

C. Miranda, Partial Differential Equations of Elliptic Type, 2nd edn., Springer, 1970.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. SladekV. 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.   Google Scholar

[27]

J. SladekV. 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.  Google Scholar

[28]

O. Steinbach, Numerical Approximation Methods for Elliptic Boundary Value Problems, Springer Berlin, 2007. doi: 10.1007/978-0-387-68805-3.  Google Scholar

[29]

A. E. Taigbenu, The Green Element Method, Kluwer Academic Publishers, Boston-Dordrecht-London, 1999. Google Scholar

[30]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.  Google Scholar

[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.  Google Scholar

[32]

T. ZhuJ.-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.  Google Scholar

[33]

T. ZhuJ.-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.   Google Scholar

show all references

References:
[1]

O. ChkaduaS. 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.  Google Scholar

[2]

O. ChkaduaS. 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.  Google Scholar

[3]

M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal., 19 (1988), 613-626.  doi: 10.1137/0519043.  Google Scholar

[4]

G. Eskin, Boundary Value Problems for Elliptic Pseudodifferential Equations, Transl. of Mathem. Monographs, Amer. Math. Soc., vol. 52: Providence, Rhode Island, 1981.  Google Scholar

[5]

R. GuttM. KohrS. 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.  Google Scholar

[6]

R. GrzhibovskisS. 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.  Google Scholar

[7]

D. Hilbert, Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen, Teubner, Leipzig-Berlin, 2nd edition, 1924.  Google Scholar

[8]

G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, Springer, Berlin, 2008. doi: 10.1007/978-3-540-68545-6.  Google Scholar

[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.  Google Scholar

[10]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon & Breach, New York, 1969.  Google Scholar

[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.   Google Scholar

[12]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer, Berlin, 1973.  Google Scholar

[13]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000.  Google Scholar

[14]

S. G. Michlin and S. Prössdorf, Singular Integral Operators, Springer Berlin, 1986. doi: 10.1007/978-3-642-61631-0.  Google Scholar

[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.  Google Scholar

[16]

S. E. Mikhailov, Localized boundary-domain integral formulations for problems with variable coefficients, Engineering Analysis with Boundary Elements, 26 (2002), 681-690.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[21]

C. Miranda, Partial Differential Equations of Elliptic Type, 2nd edn., Springer, 1970.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. SladekV. 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.   Google Scholar

[27]

J. SladekV. 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.  Google Scholar

[28]

O. Steinbach, Numerical Approximation Methods for Elliptic Boundary Value Problems, Springer Berlin, 2007. doi: 10.1007/978-0-387-68805-3.  Google Scholar

[29]

A. E. Taigbenu, The Green Element Method, Kluwer Academic Publishers, Boston-Dordrecht-London, 1999. Google Scholar

[30]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.  Google Scholar

[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.  Google Scholar

[32]

T. ZhuJ.-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.  Google Scholar

[33]

T. ZhuJ.-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.   Google Scholar

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