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October  2021, 20(10): 3561-3581. doi: 10.3934/cpaa.2021121

## 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 Published  October 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 and Applied Analysis, 2021, 20 (10) : 3561-3581. doi: 10.3934/cpaa.2021121
##### References:
 [1] Y. Achdou, O. 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. [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. [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. [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. [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. [6] G. Beavers and D. Joseph, Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30 (1967), 197-207. [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. [8] T. Clopeau, A. 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. [9] S. Ding, Q. 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. [10] L. Evans, Partial Differential Equations, Amer. Math. Soc., Providence RI, 1998. [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. [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. [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. [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. [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. [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. [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. [18] P. Lions, Mathematical Topics in Fluid Mechanics, Volume I, Incompressible Models, Oxford Science Publications, 1998. [19] J. Magnaudet, M. 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. [20] C. Navier, Sur les lois de l'équilibre et du mouvement des corps élastiques, Mem. Acad. R. Sci. Inst. France, 6 (1827), 369. [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. [22] J. Serrin, Mathematical Principles of Classical Fluid Mechanics, Encyclopedia of Physics VIII/1, Springer-Verlag, Berlin, 1959. [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. [24] R. Temam, Navier–Stokes Equations: Theory and Numerical Analysis, AMS Chelsea edition, Providence RI, 2001. doi: 10.1090/chel/343. [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. [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. [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.

show all references

##### References:
 [1] Y. Achdou, O. 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. [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. [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. [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. [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. [6] G. Beavers and D. Joseph, Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30 (1967), 197-207. [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. [8] T. Clopeau, A. 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. [9] S. Ding, Q. 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. [10] L. Evans, Partial Differential Equations, Amer. Math. Soc., Providence RI, 1998. [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. [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. [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. [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. [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. [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. [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. [18] P. Lions, Mathematical Topics in Fluid Mechanics, Volume I, Incompressible Models, Oxford Science Publications, 1998. [19] J. Magnaudet, M. 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. [20] C. Navier, Sur les lois de l'équilibre et du mouvement des corps élastiques, Mem. Acad. R. Sci. Inst. France, 6 (1827), 369. [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. [22] J. Serrin, Mathematical Principles of Classical Fluid Mechanics, Encyclopedia of Physics VIII/1, Springer-Verlag, Berlin, 1959. [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. [24] R. Temam, Navier–Stokes Equations: Theory and Numerical Analysis, AMS Chelsea edition, Providence RI, 2001. doi: 10.1090/chel/343. [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. [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. [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.
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