doi: 10.3934/cpaa.2021121
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Global well-posedness of the Navier-Stokes equations with Navier-slip boundary conditions in a strip domain

1. 

College of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060, Guangdong, China

2. 

South China Research Center for Applied Mathematics and Interdisciplinary Studies, South China Normal University, Guangzhou, 510631, Guangdong, China

* Corresponding author

Received  April 2019 Revised  June 2021 Early access July 2021

Fund Project: Li's research is supported by the National Natural Science Foundation of China(No.11901399) and the Natural Science Foundation of Shenzhen University (2019084). Ding's research is supported by the National Natural Science Foundation of China (No.11371152, No.11571117, No.11871005 and No.11771155), Natural Science Foundation of Guandong Province (No.2017A030313003 and No.2021A1515010303) and Science and Technology Program of Guangzhou (No.2019050001)

This paper is concerned with the existence and uniqueness of the strong solution to the incompressible Navier-Stokes equations with Navier-slip boundary conditions in a two-dimensional strip domain where the slip coefficients may not have defined sign. In the meantime, we also establish a number of Gagliardo-Nirenberg inequalities in the corresponding Sobolev spaces which will be applicable to other similar situations.

Citation: Quanrong Li, Shijin Ding. Global well-posedness of the Navier-Stokes equations with Navier-slip boundary conditions in a strip domain. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021121
References:
[1]

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

[2]

C. Amrouche and A. Rejaiba, $L^p$ theory for Stokes and Navier-Stokes equations with Navier boundary condition, J. Differ. Equ., 256 (2014), 1515-1547.  doi: 10.1016/j.jde.2013.11.005.  Google Scholar

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C. Amrouche and N. Seloula, On the Stokes equations with the Navier-type boundary conditions, Differ. Equ. Appl., 3 (2011), 581-607.  doi: 10.7153/dea-03-36.  Google Scholar

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E. Bänsch, Finite element discretization of the Navier-Stokes equations with free capillary surface, Numer. Math., 88 (2001), 203-235.  doi: 10.1007/PL00005443.  Google Scholar

[6]

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

[7]

D. Chauhan and K. Shekhawat, Heat transfer in Couette flow of a compressible Newtonian fluid in the presence of a naturally permeable boundary, J. Phys.D: Appl. Phys., 26 (1993), 933-936.   Google Scholar

[8]

T. ClopeauA. Mikelić and R. Robert, On the vanishing viscosity limit for the 2D incompressible Navier-Stokes equations with the friction type boundary conditions, Nonlinearity, 11 (1998), 1625-1636.  doi: 10.1088/0951-7715/11/6/011.  Google Scholar

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S. DingQ. Li and Z. Xin, Stability analysis for the incompressible Navier-Stokes equations with Navier boundary conditions, J. Math. Fluid Mech., 20 (2018), 603-629.  doi: 10.1007/s00021-017-0337-2.  Google Scholar

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L. Evans, Partial Differential Equations, Amer. Math. Soc., Providence RI, 1998.  Google Scholar

[11]

G. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equuations, Springer Monographs in Mathematics, 2$^nd$ Edition, 2011. doi: 10.1007/978-0-387-09620-9.  Google Scholar

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G. Gie and J. Kelliher, Boundary layer analysis of the Navier-Stokes equations with generalized Navier boundary conditions, J. Differ. Equ., 253 (2012), 1862-1892.  doi: 10.1016/j.jde.2012.06.008.  Google Scholar

[13]

A. Haase, J. Wood, R. Lammertink, J. Snoeijer, Why bumpy is better: the role of the dissipaption distribution in slip flow over a bubble mattress, Phys. Rev. Fluid, 1 (2016), 054101. Google Scholar

[14]

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

[15]

V. John, Slip with friction and penetration with resistance boundary conditions for the Navier-Stokes equation-numerical test and aspect of the implementation, J. Comput. Appl. Math., 147 (2002), 287-300.  doi: 10.1016/S0377-0427(02)00437-5.  Google Scholar

[16]

J. Kelliher, Navier-Stokes equations with Navier boundary conditions for a bounded domain in plane, SIAM J. Math. Anal., 38 (2006), 210-232.  doi: 10.1137/040612336.  Google Scholar

[17]

H. Li and X. Zhang, Stability of plane Couette flow for the compressible Navier-Stokes equations with Navier-slip boundary, J. Differ. Equ., 263 (2017), 1160-1187.  doi: 10.1016/j.jde.2017.03.009.  Google Scholar

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P. Lions, Mathematical Topics in Fluid Mechanics, Volume I, Incompressible Models, Oxford Science Publications, 1998.  Google Scholar

[19]

J. MagnaudetM. Riverot and J. Fabre, Accelerated flows past a rigid sphere or a spherical bubble. Part 1. Steady straining flow, J. Fluid Mech., 284 (1995), 97-135.  doi: 10.1017/S0022112095000280.  Google Scholar

[20]

C. Navier, Sur les lois de l'équilibre et du mouvement des corps élastiques, Mem. Acad. R. Sci. Inst. France, 6 (1827), 369. Google Scholar

[21]

T. Qian, X. Wang and P. Sheng, Molecular scale contact line hydrodynamics of immiscible flows, Phys. Rev. E, 68 (2003), 016306. doi: 10.1103/PhysRevE.68.016306.  Google Scholar

[22]

J. Serrin, Mathematical Principles of Classical Fluid Mechanics, Encyclopedia of Physics VIII/1, Springer-Verlag, Berlin, 1959.  Google Scholar

[23]

V. Solonnikov and V. Ščadilov, A certain boundary value problem for the stationary system of Navier-Stokes equations, Trudy Mat. Inst. Steklov., 125(1973), 196–210; translation in Proc. Steklov Inst. Math., 125 (1973), 186–199.  Google Scholar

[24]

R. Temam, Navier–Stokes Equations: Theory and Numerical Analysis, AMS Chelsea edition, Providence RI, 2001. doi: 10.1090/chel/343.  Google Scholar

[25]

H. da Veiga, On the regularity of flows with Ladyzhenskaya shear-dependent viscosity and slip or nonslip boundary conditions, Commun. Pure Appl. Math., LVIII (2005), 552-577.  doi: 10.1002/cpa.20036.  Google Scholar

[26]

Y. Xiao and Z. Xin, On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition, Commun. Pure Appl. Math., 60 (2007), 1027-1055.  doi: 10.1002/cpa.20187.  Google Scholar

[27]

Y. Xiao and Z. Xin, On the inviscid limit of the 3D Navier-Stokes equations with generalized Navier-slip boundary conditions, Commun. Math. Stat., 1 (2013), 259-279.  doi: 10.1007/s40304-013-0014-6.  Google Scholar

show all references

References:
[1]

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

[2]

C. Amrouche and A. Rejaiba, $L^p$ theory for Stokes and Navier-Stokes equations with Navier boundary condition, J. Differ. Equ., 256 (2014), 1515-1547.  doi: 10.1016/j.jde.2013.11.005.  Google Scholar

[3]

C. Amrouche and N. Seloula, On the Stokes equations with the Navier-type boundary conditions, Differ. Equ. Appl., 3 (2011), 581-607.  doi: 10.7153/dea-03-36.  Google Scholar

[4]

S. Antontsev and H. de Oliveira, Navier-Stokes equations with absorption under slip boundary conditions: existence, uniqueness and extinction in time, RIMS Kôkyûroku Bessatsu, B1 (2007), 21–41.  Google Scholar

[5]

E. Bänsch, Finite element discretization of the Navier-Stokes equations with free capillary surface, Numer. Math., 88 (2001), 203-235.  doi: 10.1007/PL00005443.  Google Scholar

[6]

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

[7]

D. Chauhan and K. Shekhawat, Heat transfer in Couette flow of a compressible Newtonian fluid in the presence of a naturally permeable boundary, J. Phys.D: Appl. Phys., 26 (1993), 933-936.   Google Scholar

[8]

T. ClopeauA. Mikelić and R. Robert, On the vanishing viscosity limit for the 2D incompressible Navier-Stokes equations with the friction type boundary conditions, Nonlinearity, 11 (1998), 1625-1636.  doi: 10.1088/0951-7715/11/6/011.  Google Scholar

[9]

S. DingQ. Li and Z. Xin, Stability analysis for the incompressible Navier-Stokes equations with Navier boundary conditions, J. Math. Fluid Mech., 20 (2018), 603-629.  doi: 10.1007/s00021-017-0337-2.  Google Scholar

[10]

L. Evans, Partial Differential Equations, Amer. Math. Soc., Providence RI, 1998.  Google Scholar

[11]

G. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equuations, Springer Monographs in Mathematics, 2$^nd$ Edition, 2011. doi: 10.1007/978-0-387-09620-9.  Google Scholar

[12]

G. Gie and J. Kelliher, Boundary layer analysis of the Navier-Stokes equations with generalized Navier boundary conditions, J. Differ. Equ., 253 (2012), 1862-1892.  doi: 10.1016/j.jde.2012.06.008.  Google Scholar

[13]

A. Haase, J. Wood, R. Lammertink, J. Snoeijer, Why bumpy is better: the role of the dissipaption distribution in slip flow over a bubble mattress, Phys. Rev. Fluid, 1 (2016), 054101. Google Scholar

[14]

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

[15]

V. John, Slip with friction and penetration with resistance boundary conditions for the Navier-Stokes equation-numerical test and aspect of the implementation, J. Comput. Appl. Math., 147 (2002), 287-300.  doi: 10.1016/S0377-0427(02)00437-5.  Google Scholar

[16]

J. Kelliher, Navier-Stokes equations with Navier boundary conditions for a bounded domain in plane, SIAM J. Math. Anal., 38 (2006), 210-232.  doi: 10.1137/040612336.  Google Scholar

[17]

H. Li and X. Zhang, Stability of plane Couette flow for the compressible Navier-Stokes equations with Navier-slip boundary, J. Differ. Equ., 263 (2017), 1160-1187.  doi: 10.1016/j.jde.2017.03.009.  Google Scholar

[18]

P. Lions, Mathematical Topics in Fluid Mechanics, Volume I, Incompressible Models, Oxford Science Publications, 1998.  Google Scholar

[19]

J. MagnaudetM. Riverot and J. Fabre, Accelerated flows past a rigid sphere or a spherical bubble. Part 1. Steady straining flow, J. Fluid Mech., 284 (1995), 97-135.  doi: 10.1017/S0022112095000280.  Google Scholar

[20]

C. Navier, Sur les lois de l'équilibre et du mouvement des corps élastiques, Mem. Acad. R. Sci. Inst. France, 6 (1827), 369. Google Scholar

[21]

T. Qian, X. Wang and P. Sheng, Molecular scale contact line hydrodynamics of immiscible flows, Phys. Rev. E, 68 (2003), 016306. doi: 10.1103/PhysRevE.68.016306.  Google Scholar

[22]

J. Serrin, Mathematical Principles of Classical Fluid Mechanics, Encyclopedia of Physics VIII/1, Springer-Verlag, Berlin, 1959.  Google Scholar

[23]

V. Solonnikov and V. Ščadilov, A certain boundary value problem for the stationary system of Navier-Stokes equations, Trudy Mat. Inst. Steklov., 125(1973), 196–210; translation in Proc. Steklov Inst. Math., 125 (1973), 186–199.  Google Scholar

[24]

R. Temam, Navier–Stokes Equations: Theory and Numerical Analysis, AMS Chelsea edition, Providence RI, 2001. doi: 10.1090/chel/343.  Google Scholar

[25]

H. da Veiga, On the regularity of flows with Ladyzhenskaya shear-dependent viscosity and slip or nonslip boundary conditions, Commun. Pure Appl. Math., LVIII (2005), 552-577.  doi: 10.1002/cpa.20036.  Google Scholar

[26]

Y. Xiao and Z. Xin, On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition, Commun. Pure Appl. Math., 60 (2007), 1027-1055.  doi: 10.1002/cpa.20187.  Google Scholar

[27]

Y. Xiao and Z. Xin, On the inviscid limit of the 3D Navier-Stokes equations with generalized Navier-slip boundary conditions, Commun. Math. Stat., 1 (2013), 259-279.  doi: 10.1007/s40304-013-0014-6.  Google Scholar

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