March  2021, 20(3): 1103-1133. doi: 10.3934/cpaa.2021009

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

1. 

Department of Mathematics, Brunel University London, UK

2. 

Department of Quantitative Methods, Universidad Loyola Andalucía, Sevilla, Spain

* Corresponding author

Received  August 2020 Revised  November 2020 Published  March 2021 Early access  January 2021

Fund Project: 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

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.

Citation: Sergey E. Mikhailov, Carlos F. Portillo. Boundary-Domain Integral Equations equivalent to an exterior mixed BVP for the variable-viscosity compressible Stokes PDEs. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1103-1133. doi: 10.3934/cpaa.2021009
References:
[1]

F. Alliot and C. Amrouche, Weak solutions for the exterior Stokes problem in weighted Sobolev spaces, Math. Meth. Appl. Sci., 23 (2000), 575-600.  doi: 10.1002/(SICI)1099-1476(200004)23:6<575::AID-MMA128>3.0.CO;2-4.

[2]

C. AmroucheV. Girault and J. Giroire, Dirichlet and Neumann exterior problems for the $n$-dimensional Laplace operator. An approach in weighted Sobolev spaces, J. Math. Pures Appl., 76 (1997), 55-81.  doi: 10.1016/S0021-7824(97)89945-X.

[3]

A. Bossavit, Électromagnétisme, en Vue de la Modélisation, Springer, Berlin, 2003.

[4]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.

[5]

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 Equ. Appl., 21 (2009), 499-543.  doi: 10.1216/JIE-2009-21-4-499.

[6]

O. Chkadua, S. E. Mikhailov and D. Natroshvili, Analysis of direct segregated boundary-domain integral equations for variable-coefficient mixed BVPs in exterior domains, Anal. Appl., 11 (2013), 1350006, 33 pp. doi: 10.1142/S0219530513500061.

[7]

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

[8]

T. T. Dufera and S. E. Mikhailov, Boundary-domain integral equations for variable-coefficient Dirichlet BVP in 2D unbounded domain, in Analysis, Probability, Applications, and Computations (eds. Lindahl et al), Springer Nature, Switzerland AG, 2019. doi: 10.1007/978-3-030-04459-6_46.

[9]

C. Fresneda-Portillo and S. E. Mikhailov, Analysis of boundary-domain integral equations to the mixed BVP for a compressible Stokes system with variable viscosity, Commun. Pure Appl. Anal., 18 (2019), 3059-3088.  doi: 10.3934/cpaa.2019137.

[10]

J. Giroire, Étude de quelques problèmes aux limites extérieurs et résolution par équations intégrales, Thése de Doctorat d'État, Université Pierre-et-Marie-Curie (Paris VI), 1987.

[11]

J. Giroire and J. Nedelec, Numerical solution of an exterior Neumann problem using a double layer potential, Math. Comp., 32 (1978), 973-990.  doi: 10.2307/2006329.

[12]

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.

[13]

B. Hanouzet, Espaces de Sobolev avec Poids. Application au probleme de Dirichlet dans un demi espace, Rendiconti del Seminario Matematico della Universita di Padova, 46 (1971), 227-272. 

[14]

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

[15]

M. Kohr, M. Lanza de Cristoforis, S. E. Mikhailov and W. L. Wendland, Integral potential method for transmission problem with Lipschitz interface in $\mathbb R^3$ for the Stokes and Darcy-Forchheimer-Brinkman PDE systems, Z. Angew. Math. Phys., 67 (2016), 116, 30pp. doi: 10.1007/s00033-016-0696-1.

[16]

M. Kohr, S. E. Mikhailov and W. L. Wendland, Variational approach for layer potentials of the Stokes system with $L_{\infty}$ symmetrically elliptic coefficient tensor and applications to Stokes and Navier-Stokes boundary problems, arXiv: 2002.09990.

[17]

M. Kohr and W. L. Wendland, Variational boundary integral equations for the Stokes system, Appl. Anal., 85 (2006), 1343-1372.  doi: 10.1080/00036810600963961.

[18]

M. Kohr and W. L. Wendland, Boundary value problems for the Brinkman system with $L_{\infty }$ coefficients in Lipschitz domains on compact Riemannian manifolds. A variational approach, J. Math. Pures Appl., 131 (2019), 17-63.  doi: 10.1016/j.matpur.2019.04.002.

[19]

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

[20]

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

[21] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, UK, 2000. 
[22]

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.

[23]

S. E. Mikhailov, Localized boundary-domain integral formulations for problems with variable coefficients, Eng. Anal. Bound. Elem., 26 (2002), 681-690.  doi: 10.1016/S0955-7997(02)00030-9.

[24]

S. E. Mikhailov, Analysis of segregated boundary-domain integral equations for BVPs with non-smooth coefficient on Lipschitz domains, Bound. Value Probl., 87 (2018), 1-52.  doi: 10.1186/s13661-018-0992-0.

[25]

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 (eds. C. Constanda and A. Kirsh), Springer, Boston, 2015.

[26]

S. E. Mikhailov and C. F. Portillo, A new family of boundary-domain integral equations for a mixed elliptic BVP with variable coefficient, in Proceedings of the 10th UK Conference on Boundary Integral Methods (ed. P. Harris), Brighton University Press, 2015.

[27]

S. E. Mikhailov and C. F. Portillo, BDIEs for the compressible Stokes system with variable viscosity mixed BVP in bounded domains, in Proceedings of the 11th UK Conference on Boundary Integral Methods (ed. D.J. Chappell), Nottingham Trent Univ., 2017.

[28]

I. Mitrea and M. Mitrea, The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in non-smooth domains, T. Am. Math. Soc., 359 (2007), 4143-4182. doi: 10.1090/S0002-9947-07-04146-3.

[29]

J. Nedelec, Acoustic and Electromagnetic Equations, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4393-7.

[30]

J. Nedelec and J. Planchard, Une méthode variationnelle d'éléments finis pour la résolution numérique d'un problème extérieur dans $\mathbb{R}^{3}$, RAIRO, 7 (1973), 105-129. 

[31]

J. B. Neto, Inhomogeneous boundary value problems in a half space, Ann. Sc. Sup. Pisa, 19 (1965), 331-365. 

[32]

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.

[33]

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

[34]

R. Temam, Navier-Stokes Equations, AMS Chelsea Edition, American Mathematical Society, 2001. doi: 10.1090/chel/343.

[35]

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

[36]

W. Varnhorn, The Stokes Equations, Akademie Verlag, Berlin, 1994.

[37]

W. L. Wendland and J. Zhu, The boundary element method for three dimensional Stokes flow exterior to an open surface, Math. Comput. Model., 6 (1991), 19-42.  doi: 10.1016/0895-7177(91)90021-X.

show all references

References:
[1]

F. Alliot and C. Amrouche, Weak solutions for the exterior Stokes problem in weighted Sobolev spaces, Math. Meth. Appl. Sci., 23 (2000), 575-600.  doi: 10.1002/(SICI)1099-1476(200004)23:6<575::AID-MMA128>3.0.CO;2-4.

[2]

C. AmroucheV. Girault and J. Giroire, Dirichlet and Neumann exterior problems for the $n$-dimensional Laplace operator. An approach in weighted Sobolev spaces, J. Math. Pures Appl., 76 (1997), 55-81.  doi: 10.1016/S0021-7824(97)89945-X.

[3]

A. Bossavit, Électromagnétisme, en Vue de la Modélisation, Springer, Berlin, 2003.

[4]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.

[5]

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 Equ. Appl., 21 (2009), 499-543.  doi: 10.1216/JIE-2009-21-4-499.

[6]

O. Chkadua, S. E. Mikhailov and D. Natroshvili, Analysis of direct segregated boundary-domain integral equations for variable-coefficient mixed BVPs in exterior domains, Anal. Appl., 11 (2013), 1350006, 33 pp. doi: 10.1142/S0219530513500061.

[7]

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

[8]

T. T. Dufera and S. E. Mikhailov, Boundary-domain integral equations for variable-coefficient Dirichlet BVP in 2D unbounded domain, in Analysis, Probability, Applications, and Computations (eds. Lindahl et al), Springer Nature, Switzerland AG, 2019. doi: 10.1007/978-3-030-04459-6_46.

[9]

C. Fresneda-Portillo and S. E. Mikhailov, Analysis of boundary-domain integral equations to the mixed BVP for a compressible Stokes system with variable viscosity, Commun. Pure Appl. Anal., 18 (2019), 3059-3088.  doi: 10.3934/cpaa.2019137.

[10]

J. Giroire, Étude de quelques problèmes aux limites extérieurs et résolution par équations intégrales, Thése de Doctorat d'État, Université Pierre-et-Marie-Curie (Paris VI), 1987.

[11]

J. Giroire and J. Nedelec, Numerical solution of an exterior Neumann problem using a double layer potential, Math. Comp., 32 (1978), 973-990.  doi: 10.2307/2006329.

[12]

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.

[13]

B. Hanouzet, Espaces de Sobolev avec Poids. Application au probleme de Dirichlet dans un demi espace, Rendiconti del Seminario Matematico della Universita di Padova, 46 (1971), 227-272. 

[14]

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

[15]

M. Kohr, M. Lanza de Cristoforis, S. E. Mikhailov and W. L. Wendland, Integral potential method for transmission problem with Lipschitz interface in $\mathbb R^3$ for the Stokes and Darcy-Forchheimer-Brinkman PDE systems, Z. Angew. Math. Phys., 67 (2016), 116, 30pp. doi: 10.1007/s00033-016-0696-1.

[16]

M. Kohr, S. E. Mikhailov and W. L. Wendland, Variational approach for layer potentials of the Stokes system with $L_{\infty}$ symmetrically elliptic coefficient tensor and applications to Stokes and Navier-Stokes boundary problems, arXiv: 2002.09990.

[17]

M. Kohr and W. L. Wendland, Variational boundary integral equations for the Stokes system, Appl. Anal., 85 (2006), 1343-1372.  doi: 10.1080/00036810600963961.

[18]

M. Kohr and W. L. Wendland, Boundary value problems for the Brinkman system with $L_{\infty }$ coefficients in Lipschitz domains on compact Riemannian manifolds. A variational approach, J. Math. Pures Appl., 131 (2019), 17-63.  doi: 10.1016/j.matpur.2019.04.002.

[19]

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

[20]

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

[21] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, UK, 2000. 
[22]

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.

[23]

S. E. Mikhailov, Localized boundary-domain integral formulations for problems with variable coefficients, Eng. Anal. Bound. Elem., 26 (2002), 681-690.  doi: 10.1016/S0955-7997(02)00030-9.

[24]

S. E. Mikhailov, Analysis of segregated boundary-domain integral equations for BVPs with non-smooth coefficient on Lipschitz domains, Bound. Value Probl., 87 (2018), 1-52.  doi: 10.1186/s13661-018-0992-0.

[25]

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 (eds. C. Constanda and A. Kirsh), Springer, Boston, 2015.

[26]

S. E. Mikhailov and C. F. Portillo, A new family of boundary-domain integral equations for a mixed elliptic BVP with variable coefficient, in Proceedings of the 10th UK Conference on Boundary Integral Methods (ed. P. Harris), Brighton University Press, 2015.

[27]

S. E. Mikhailov and C. F. Portillo, BDIEs for the compressible Stokes system with variable viscosity mixed BVP in bounded domains, in Proceedings of the 11th UK Conference on Boundary Integral Methods (ed. D.J. Chappell), Nottingham Trent Univ., 2017.

[28]

I. Mitrea and M. Mitrea, The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in non-smooth domains, T. Am. Math. Soc., 359 (2007), 4143-4182. doi: 10.1090/S0002-9947-07-04146-3.

[29]

J. Nedelec, Acoustic and Electromagnetic Equations, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4393-7.

[30]

J. Nedelec and J. Planchard, Une méthode variationnelle d'éléments finis pour la résolution numérique d'un problème extérieur dans $\mathbb{R}^{3}$, RAIRO, 7 (1973), 105-129. 

[31]

J. B. Neto, Inhomogeneous boundary value problems in a half space, Ann. Sc. Sup. Pisa, 19 (1965), 331-365. 

[32]

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.

[33]

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

[34]

R. Temam, Navier-Stokes Equations, AMS Chelsea Edition, American Mathematical Society, 2001. doi: 10.1090/chel/343.

[35]

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

[36]

W. Varnhorn, The Stokes Equations, Akademie Verlag, Berlin, 1994.

[37]

W. L. Wendland and J. Zhu, The boundary element method for three dimensional Stokes flow exterior to an open surface, Math. Comput. Model., 6 (1991), 19-42.  doi: 10.1016/0895-7177(91)90021-X.

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