June  2022, 15(6): 1403-1420. doi: 10.3934/dcdss.2022086

$ L^p $-strong solution for the stationary exterior Stokes equations with Navier boundary condition

LR Analysis and Control of PDEs, LR22ES03, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, Tunisia

Received  December 2021 Revised  March 2022 Published  June 2022 Early access  April 2022

This paper treats the stationary Stokes problem in exterior domain of $ {{\mathbb{R}}}^3 $ with Navier slip boundary condition. The behavior at infinity of the data and the solution are determined by setting the problem in $ L^p $-spaces, for $ p> 2 $, with weights. The main results are the existence and uniqueness of strong solutions of the corresponding system.

Citation: Anis Dhifaoui. $ L^p $-strong solution for the stationary exterior Stokes equations with Navier boundary condition. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1403-1420. doi: 10.3934/dcdss.2022086
References:
[1]

P. AcevedoC. AmroucheC. Conca and A. Ghosh, Stokes and Navier-Stokes equations with Navier boundary condition, C. R. Math. Acad. Sci. Paris, 357 (2019), 115-119.  doi: 10.1016/j.crma.2018.12.002.

[2]

Y. Achdou and O. Pironneau, Domain decomposition and wall laws, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 541-547.  doi: 10.1016/S0045-7825(97)00118-7.

[3]

Y. AchdouO. Pironneau and F. Valentin, Effective boundary conditions for laminar flows over periodic rough boundaries, J. Comput. Phys., 147 (1998), 187-218.  doi: 10.1006/jcph.1998.6088.

[4]

Y. Achdou, O. Pironneau and F. Valentin, Shape control versus boundary control, Équations aux Dérivées Partielles et Applications, Gauthier-Villars, Éd. Sci. Méd. Elsevier, Paris, (1998), 1–18.

[5]

F. Alliot and C. Amrouche, The Stokes problem in $\mathbb{R}^n$: An approach in weighted Sobolev spaces, Math. Models Methods Appl. Sci., 9 (1999), 723-754.  doi: 10.1142/S0218202599000361.

[6]

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

[7]

C. Amrouche and V. Girault, Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. J., 44 (1994), 109-140. 

[8]

C. AmroucheV. Girault and J. Giroire, Weighted Sobolev spaces for Laplace's equation in $\mathbb{ R}^n$, J. Math. Pures Appl., 73 (1994), 579-606. 

[9]

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.

[10]

C. Amrouche and M. Meslameni, Stokes problem with several types of boundary conditions in an exterior domain, Electron. J. Differential Equations, (2013), No. 196, 28 pp.

[11]

C. Amrouche and A. Rejaiba, $\mathit{\boldsymbol{L}}^p$-theory for Stokes and Navier-Stokes equations with Navier boundary condition, J. Differential Equations, 256 (2014), 1515-1547.  doi: 10.1016/j.jde.2013.11.005.

[12]

C. Amrouche and A. Rejaiba, Navier-Stokes equations with Navier boundary condition, Math. Methods Appl. Sci., 39 (2016), 5091-5112.  doi: 10.1002/mma.3338.

[13]

S. N. Antontsev and H. B. de Oliveira, Navier-Stokes equations with absorption under slip boundary conditions: Existence, uniqueness and extinction in time, Kyoto Conference on the Navier-Stokes Equations and their Applications, RIMS Kôkyûroku Bessatsu, B1, Res. Inst. Math. Sci. (RIMS), Kyoto, (2007), 21-41. 

[14]

A. Basson and D. Gérard-Varet, Wall laws for fluid flows at a boundary with random roughness, Comm. Pure Appl. Math., 61 (2008), 941-987.  doi: 10.1002/cpa.20237.

[15]

H. Beirão Da Veiga, Regularity for Stokes and generalized Stokes systems under nonhomogeneous slip-type boundary conditions, Adv. Differential Equations, 9 (2014), 1079-1114. 

[16]

H. Beirão da Veiga, Vorticity and regularity for flows under the Navier boundary condition, Commun. Pure Appl. Anal., 5 (2006), 907-918.  doi: 10.3934/cpaa.2006.5.907.

[17]

J. Casado-DíazE. Fernández-Cara and J. Simon, Why viscous fluids adhere to rugose walls: A mathematical explanation, J. Differential Equations, 189 (2003), 526-537.  doi: 10.1016/S0022-0396(02)00115-8.

[18]

A. Dhifaoui, Équations de Stokes en Domaine Extérieur Avec des Conditions aux Limites de Type Navier, PhD thesis, 2020. Thèse de doctorat dirigée par Razafison, Ulrich Jerry et Ben Hamed, Bassem Mathématiques Bourgogne Franche-Comté 2020.

[19]

A. Dhifaoui, $ {L}^{p} $-theory for the exterior Stokes problem with Navier's type slip-without-friction boundary conditions, preprint, (2021), arXiv: 2111.05822.

[20]

A. Dhifaoui, Very weak solution for the exterior stationary Stokes equations with Navier slip boundary condition, preprint, (2021), arXiv: 2111.05824.

[21]

A. DhifaouiM. Meslameni and U. Razafison, Weighted Hilbert spaces for the stationary exterior Stokes problem with Navier slip boundary conditions, J. Math. Anal. Appl., 472 (2019), 1846-1871.  doi: 10.1016/j.jmaa.2018.12.026.

[22]

E. Friedmann, The optimal shape of riblets in the viscous sublayer, J. Math. Fluid Mech., 12 (2010), 243-265.  doi: 10.1007/s00021-008-0284-z.

[23]

E. Friedmann and T. Richter, Optimal microstructures drag reducing mechanism of riblets, J. Math. Fluid Mech., 13 (2011), 429-447.  doi: 10.1007/s00021-010-0033-y.

[24]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I. Linearized Steady Problems, Springer Tracts in Natural Philosophy, 38. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.

[25]

D. Gérard-Varet and N. Masmoudi, Relevance of the slip condition for fluid flows near an irregular boundary, Comm. Math. Phys., 295 (2010), 99-137.  doi: 10.1007/s00220-009-0976-0.

[26]

V. Girault, The Stokes problem and vector potential operator in three-dimensional exterior domains: An approach in weighted Sobolev spaces, Differential Integral Equations, 7 (1994), 535-570. 

[27]

V. GiraultJ. Giroire and A. Sequeira, A stream-function–vorticity variational formulation for the exterior Stokes problem in weighted Sobolev spaces, Math. Methods Appl. Sci., 15 (1992), 345-363.  doi: 10.1002/mma.1670150506.

[28]

V. Girault and A. Sequeira, A well-posed problem for the exterior Stokes equations in two and three dimensions, Arch. Rational Mech. Anal., 114 (1991), 313-333.  doi: 10.1007/BF00376137.

[29]

J. Giroire, Étude de Quelques Problèmes aux Limites Extérieurs et Résolution Par équations Intégrales, Thèse de Doctorat d'État, Université Pierre et Marie Curie, Paris, 1987.

[30]

B. Hanouzet, Espaces de Sobolev avec poids application au problème de Dirichlet dans un demi espace, Rend. Sem. Mat. Univ. Padova, 46 (1971), 227-272. 

[31]

W. Jäger and A. Mikelić, On the roughness-induced effective boundary conditions for an incompressible viscous flow, J. Differential Equations, 170 (2001), 96-122.  doi: 10.1006/jdeq.2000.3814.

[32]

D. D. Joseph and G. S. Beavers, Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30 (1967), 197-207. 

[33]

H. LouatiM. Meslameni and U. Razafison, Weighted $L^p$-theory for vector potential operators in three-dimensional exterior domains, Math. Methods Appl. Sci., 39 (2016), 1990-2010.  doi: 10.1002/mma.3615.

[34]

H. Louati, M. Meslameni and U. Razafison, On the three-dimensional stationary exterior stokes problem with non standard boundary conditions, ZAMM Z. Angew. Math. Mech., 100 (2020), e201900181, 28 pp. doi: 10.1002/zamm.201900181.

[35]

G. Mulone and F. Salemi, On the existence of hydrodynamic motion in a domain with free boundary type conditions, Meccanica, 18 (1983), 136-144. 

[36]

G. Mulone and F. Salemi, On the hydrodynamic motion in a domain with mixed boundary conditions: Existence, uniqueness, stability and linearization principle, Ann. Mat. Pura Appl., 139 (1985), 147-174.  doi: 10.1007/BF01766853.

[37]

C. L. M. H. Navier, Mémoire sur les Lois du Mouvement des fluides, Mem. Acad. Sci. Inst. de France, 6 (1827), 389-440. 

[38]

A. Russo and A. Tartaglione, On the Navier problem for the stationary Navier-Stokes equations, J. Differential Equations, 251 (2011), 2387-2408.  doi: 10.1016/j.jde.2011.07.001.

[39]

V. A. Solonnikov and V. E. Ščadilov, A certain boundary value problem for the stationary system of Navier-Stokes equations, Trudy Mat. Int. Steklov., 125 (1973), 196-210,235. 

[40]

M. Specovius-Neugebauer, Exterior Stokes problems and decay at infinity, Math. Methods Appl. Sci., 8 (1986), 351-367.  doi: 10.1002/mma.1670080124.

[41]

M. Specovius-Neugebauer, Weak solutions of the Stokes problem in weighted Sobolev spaces, Acta Appl. Math., 37 (1994), 195-203.  doi: 10.1007/BF00995141.

show all references

References:
[1]

P. AcevedoC. AmroucheC. Conca and A. Ghosh, Stokes and Navier-Stokes equations with Navier boundary condition, C. R. Math. Acad. Sci. Paris, 357 (2019), 115-119.  doi: 10.1016/j.crma.2018.12.002.

[2]

Y. Achdou and O. Pironneau, Domain decomposition and wall laws, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 541-547.  doi: 10.1016/S0045-7825(97)00118-7.

[3]

Y. AchdouO. Pironneau and F. Valentin, Effective boundary conditions for laminar flows over periodic rough boundaries, J. Comput. Phys., 147 (1998), 187-218.  doi: 10.1006/jcph.1998.6088.

[4]

Y. Achdou, O. Pironneau and F. Valentin, Shape control versus boundary control, Équations aux Dérivées Partielles et Applications, Gauthier-Villars, Éd. Sci. Méd. Elsevier, Paris, (1998), 1–18.

[5]

F. Alliot and C. Amrouche, The Stokes problem in $\mathbb{R}^n$: An approach in weighted Sobolev spaces, Math. Models Methods Appl. Sci., 9 (1999), 723-754.  doi: 10.1142/S0218202599000361.

[6]

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

[7]

C. Amrouche and V. Girault, Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. J., 44 (1994), 109-140. 

[8]

C. AmroucheV. Girault and J. Giroire, Weighted Sobolev spaces for Laplace's equation in $\mathbb{ R}^n$, J. Math. Pures Appl., 73 (1994), 579-606. 

[9]

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.

[10]

C. Amrouche and M. Meslameni, Stokes problem with several types of boundary conditions in an exterior domain, Electron. J. Differential Equations, (2013), No. 196, 28 pp.

[11]

C. Amrouche and A. Rejaiba, $\mathit{\boldsymbol{L}}^p$-theory for Stokes and Navier-Stokes equations with Navier boundary condition, J. Differential Equations, 256 (2014), 1515-1547.  doi: 10.1016/j.jde.2013.11.005.

[12]

C. Amrouche and A. Rejaiba, Navier-Stokes equations with Navier boundary condition, Math. Methods Appl. Sci., 39 (2016), 5091-5112.  doi: 10.1002/mma.3338.

[13]

S. N. Antontsev and H. B. de Oliveira, Navier-Stokes equations with absorption under slip boundary conditions: Existence, uniqueness and extinction in time, Kyoto Conference on the Navier-Stokes Equations and their Applications, RIMS Kôkyûroku Bessatsu, B1, Res. Inst. Math. Sci. (RIMS), Kyoto, (2007), 21-41. 

[14]

A. Basson and D. Gérard-Varet, Wall laws for fluid flows at a boundary with random roughness, Comm. Pure Appl. Math., 61 (2008), 941-987.  doi: 10.1002/cpa.20237.

[15]

H. Beirão Da Veiga, Regularity for Stokes and generalized Stokes systems under nonhomogeneous slip-type boundary conditions, Adv. Differential Equations, 9 (2014), 1079-1114. 

[16]

H. Beirão da Veiga, Vorticity and regularity for flows under the Navier boundary condition, Commun. Pure Appl. Anal., 5 (2006), 907-918.  doi: 10.3934/cpaa.2006.5.907.

[17]

J. Casado-DíazE. Fernández-Cara and J. Simon, Why viscous fluids adhere to rugose walls: A mathematical explanation, J. Differential Equations, 189 (2003), 526-537.  doi: 10.1016/S0022-0396(02)00115-8.

[18]

A. Dhifaoui, Équations de Stokes en Domaine Extérieur Avec des Conditions aux Limites de Type Navier, PhD thesis, 2020. Thèse de doctorat dirigée par Razafison, Ulrich Jerry et Ben Hamed, Bassem Mathématiques Bourgogne Franche-Comté 2020.

[19]

A. Dhifaoui, $ {L}^{p} $-theory for the exterior Stokes problem with Navier's type slip-without-friction boundary conditions, preprint, (2021), arXiv: 2111.05822.

[20]

A. Dhifaoui, Very weak solution for the exterior stationary Stokes equations with Navier slip boundary condition, preprint, (2021), arXiv: 2111.05824.

[21]

A. DhifaouiM. Meslameni and U. Razafison, Weighted Hilbert spaces for the stationary exterior Stokes problem with Navier slip boundary conditions, J. Math. Anal. Appl., 472 (2019), 1846-1871.  doi: 10.1016/j.jmaa.2018.12.026.

[22]

E. Friedmann, The optimal shape of riblets in the viscous sublayer, J. Math. Fluid Mech., 12 (2010), 243-265.  doi: 10.1007/s00021-008-0284-z.

[23]

E. Friedmann and T. Richter, Optimal microstructures drag reducing mechanism of riblets, J. Math. Fluid Mech., 13 (2011), 429-447.  doi: 10.1007/s00021-010-0033-y.

[24]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I. Linearized Steady Problems, Springer Tracts in Natural Philosophy, 38. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.

[25]

D. Gérard-Varet and N. Masmoudi, Relevance of the slip condition for fluid flows near an irregular boundary, Comm. Math. Phys., 295 (2010), 99-137.  doi: 10.1007/s00220-009-0976-0.

[26]

V. Girault, The Stokes problem and vector potential operator in three-dimensional exterior domains: An approach in weighted Sobolev spaces, Differential Integral Equations, 7 (1994), 535-570. 

[27]

V. GiraultJ. Giroire and A. Sequeira, A stream-function–vorticity variational formulation for the exterior Stokes problem in weighted Sobolev spaces, Math. Methods Appl. Sci., 15 (1992), 345-363.  doi: 10.1002/mma.1670150506.

[28]

V. Girault and A. Sequeira, A well-posed problem for the exterior Stokes equations in two and three dimensions, Arch. Rational Mech. Anal., 114 (1991), 313-333.  doi: 10.1007/BF00376137.

[29]

J. Giroire, Étude de Quelques Problèmes aux Limites Extérieurs et Résolution Par équations Intégrales, Thèse de Doctorat d'État, Université Pierre et Marie Curie, Paris, 1987.

[30]

B. Hanouzet, Espaces de Sobolev avec poids application au problème de Dirichlet dans un demi espace, Rend. Sem. Mat. Univ. Padova, 46 (1971), 227-272. 

[31]

W. Jäger and A. Mikelić, On the roughness-induced effective boundary conditions for an incompressible viscous flow, J. Differential Equations, 170 (2001), 96-122.  doi: 10.1006/jdeq.2000.3814.

[32]

D. D. Joseph and G. S. Beavers, Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30 (1967), 197-207. 

[33]

H. LouatiM. Meslameni and U. Razafison, Weighted $L^p$-theory for vector potential operators in three-dimensional exterior domains, Math. Methods Appl. Sci., 39 (2016), 1990-2010.  doi: 10.1002/mma.3615.

[34]

H. Louati, M. Meslameni and U. Razafison, On the three-dimensional stationary exterior stokes problem with non standard boundary conditions, ZAMM Z. Angew. Math. Mech., 100 (2020), e201900181, 28 pp. doi: 10.1002/zamm.201900181.

[35]

G. Mulone and F. Salemi, On the existence of hydrodynamic motion in a domain with free boundary type conditions, Meccanica, 18 (1983), 136-144. 

[36]

G. Mulone and F. Salemi, On the hydrodynamic motion in a domain with mixed boundary conditions: Existence, uniqueness, stability and linearization principle, Ann. Mat. Pura Appl., 139 (1985), 147-174.  doi: 10.1007/BF01766853.

[37]

C. L. M. H. Navier, Mémoire sur les Lois du Mouvement des fluides, Mem. Acad. Sci. Inst. de France, 6 (1827), 389-440. 

[38]

A. Russo and A. Tartaglione, On the Navier problem for the stationary Navier-Stokes equations, J. Differential Equations, 251 (2011), 2387-2408.  doi: 10.1016/j.jde.2011.07.001.

[39]

V. A. Solonnikov and V. E. Ščadilov, A certain boundary value problem for the stationary system of Navier-Stokes equations, Trudy Mat. Int. Steklov., 125 (1973), 196-210,235. 

[40]

M. Specovius-Neugebauer, Exterior Stokes problems and decay at infinity, Math. Methods Appl. Sci., 8 (1986), 351-367.  doi: 10.1002/mma.1670080124.

[41]

M. Specovius-Neugebauer, Weak solutions of the Stokes problem in weighted Sobolev spaces, Acta Appl. Math., 37 (1994), 195-203.  doi: 10.1007/BF00995141.

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