# American Institute of Mathematical Sciences

August  2020, 40(8): 4579-4596. doi: 10.3934/dcds.2020193

## Boundary layer for 3D plane parallel channel flows of nonhomogeneous incompressible Navier-Stokes equations

 1 South China Research Center for Applied Mathematics and Interdisciplinary Studies, South China Normal University, Guangzhou 510631, China 2 School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China 3 School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

* Corresponding author: Zhilin Lin

Received  November 2018 Revised  October 2019 Published  May 2020

Fund Project: Ding's research is supported by the National Natural Science Foundation of China (No.11371152, No.11571117, No.11871005 and No.11771155) and Guangdong Provincial Natural Science Foundation (No.2017A030313003). Lin's research is supported by the Innovation Project of Graduate School of South China Normal University (No.2018LKXM009). Niu's research is supported by the National Natural Science Foundation of China (No.11471220 and No.11871046) while she was visiting at the Institute of Mathematical Sciences of the Chinese University of Hong Kong

In this paper, we establish the mathematical validity of the Prandtl boundary layer theory for a class of nonlinear plane parallel flows of nonhomogeneous incompressible Navier-Stokes equations. The convergence is shown under various Sobolev norms, including the physically important space-time uniform norm, as well as the $L^\infty(H^1)$ norm. It is mentioned that the mathematical validity of the Prandtl boundary layer theory for nonlinear plane parallel flow is generalized to the nonhomogeneous case.

Citation: Shijin Ding, Zhilin Lin, Dongjuan Niu. Boundary layer for 3D plane parallel channel flows of nonhomogeneous incompressible Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4579-4596. doi: 10.3934/dcds.2020193
##### References:
 [1] R. Alexander, Y.-G. Wang, C.-J. Xu and T. Yang, Well posedness of the Prandtl equation in Sobolev spaces, J. Amer. Math. Soc., 28 (2015), 745-784.  doi: 10.1090/S0894-0347-2014-00813-4. [2] C. Bardos, M. C. Lopes Filho, D. Niu, H. J. Nussenzveig Lopes and E. S. Titi, Stability of two-dimensional viscous incompressible flows under three-dimensional perturbations and inviscid symmetry breaking, SIAM J. Math. Anal., 45 (2013), 1871-1885.  doi: 10.1137/120862569. [3] H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids, Comm. Partial Differential Equations, 28 (2003), 1183-1201.  doi: 10.1081/PDE-120021191. [4] M. Fei, T. Tao and Z. Zhang, On the zero-viscosity limit of the Navier-Stokes equations in $\mathbb{R}^3_+$ without analyticity, J. Math. Pures Appl. (9) 112 (2018), 170–229. doi: 10.1016/j.matpur.2017.09.007. [5] D. Gérard-Varet and E. Dormy, On the ill-posedness of the Prandtl equations, J. Amer. Math. Soc., 23 (2010), 591-609.  doi: 10.1090/S0894-0347-09-00652-3. [6] D. Gérard-Varet and Y. Maekawa, Sobolev stability of Prandtl expansions for the steady Navier-Stokes equations, Arch. Ration. Mech. Anal., 233 (2019), 1319-1382.  doi: 10.1007/s00205-019-01380-x. [7] G.-M. Gie, J. P. Kelliher, M. C. Lopes Filho, A. L. Mazzucato and H. J. Nussenzveig Lopes, The vanishing viscosity limit for some symmetric flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2019), 1237-1280.  doi: 10.1016/j.anihpc.2018.11.006. [8] E. Grenier, On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math., 53 (2000), 1067-1091.  doi: 10.1002/1097-0312(200009)53:9<1067::AID-CPA1>3.0.CO;2-Q. [9] Y. Guo and S. Iyer, Validity of steady Prandtl layer expansions, preprint, arXiv: 1805.05891. [10] Y. Guo and T. Nguyen, Prandtl boundary layer expansions of steady Navier-Stokes flows over a moving plate, Ann. PDE, 3 (2017), 58pp. doi: 10.1007/s40818-016-0020-6. [11] D. Han, A. L. Mazzucato, D. Niu and X. Wang, Boundary layer for a class of nonlinear pipe flow, J. Differential Equations, 252 (2012), 6387-6413.  doi: 10.1016/j.jde.2012.02.012. [12] S. Iyer, Steady Prandtl boundary layer expansions over a rotating disk, Arch. Ration. Mech. Anal., 224 (2017), 421-469.  doi: 10.1007/s00205-017-1080-9. [13] J. Kim, Weak solutions of an initial boundary value problem for an incompressible viscous fluid with nonnegative density, SIAM J. Math. Anal., 18 (1987), 89-96.  doi: 10.1137/0518007. [14] C. Liu, Y. Wang and T. Yang, A well-posedness theory for the Prandtl equations in three space variables, Adv. Math., 308 (2017), 1074-1126.  doi: 10.1016/j.aim.2016.12.025. [15] C. Liu, Y. Wang and T. Yang, On the ill-posedness of the Prandtl equations in three-dimensional space, Arch. Ration. Mech. Anal., 220 (2016), 83-108.  doi: 10.1007/s00205-015-0927-1. [16] C. Liu, F. Xie and T. Yang, MHD boundary layers theory in Sobolev spaces without monotonicity. I: Well-posedness theory, Comm. Pure Appl. Math., 72 (2019), 63-121.  doi: 10.1002/cpa.21763. [17] C. Liu, F. Xie and T. Yang, Justification of Prandtl ansatz for MHD boundary layer, SIAM J. Math. Anal., 51 (2019), 2748-2791.  doi: 10.1137/18M1219618. [18] Y. Maekawa, On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half-plane, Comm. Pure Appl. Math., 67 (2014), 1045-1128.  doi: 10.1002/cpa.21516. [19] N. Masmoudi and T. K. Wong, Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods, Comm. Pure Appl. Math., 68 (2015), 1683-1741.  doi: 10.1002/cpa.21595. [20] J. E. Marsden, Well-posedness of the equations of a non-homogeneous perfect fluid, Comm. Partial Differential Equations, 1 (1976), 215-230.  doi: 10.1080/03605307608820010. [21] A. Mazzucato, D. Niu and X. Wang, Boundary layer associated with a class of 3D nonlinear plane parallel channel fows, Indiana Univ. Math. J., 60 (2011), 1113-1136.  doi: 10.1512/iumj.2011.60.4479. [22] O. A. Oleinik, On the system of Prandtl equations in boundary-layer theory, Dokl. Akad. Nauk SSR, 150 (1963), 28-31. [23] O. A. Oleinik and V. N. Samokhin, Mathematical Models in Boundary Layer Theory, Applied Mathematics and Mathematical Computation, 15, Chapman & Hall/CRC, Boca Raton, FL, 1999. [24] L. Prandtl, Über flüssigkeitsbewegungen bei sehr kleiner reibung, Verhaldlg III Int. Math. Kong, (1905), 484–491. [25] M. Sammartino and R. E. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space, I. Existence for Euler and Prandtl equations, Comm. Math. Phys., 192 (1998), 433-461.  doi: 10.1007/s002200050304. [26] M. Sammartino and R. E. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half space, Ⅱ. Construction of the Navier-Stokes solution, Comm. Math. Phys., 192 (1998), 463-491.  doi: 10.1007/s002200050305. [27] J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of viscosity, density and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117.  doi: 10.1137/0521061. [28] R. Temam and X. Wang, Asymptotic analysis of Oseen type equations in a channel at small viscosity, Indiana Univ. Math. J., 45 (1996), 863-916.  doi: 10.1512/iumj.1996.45.1290. [29] R. Temam and X. Wang, Boundary layers associated with incompressible Navier-Stokes equations: The noncharacteristic boundary case, J. Differential Equations, 179 (2002), 647-686.  doi: 10.1006/jdeq.2001.4038. [30] H. B. da Veiga and A. Valli, On the Euler equations for nonhomogeneous fluids. (Ⅰ), Rend. Sem. Mat. Univ. Padova, 63 (1980), 151-168. [31] H. B. da Veiga and A. Valli, On the Euler equations for nonhomogeneous fluids (Ⅱ), J. Math. Anal. Appl., 73 (1980), 338-350.  doi: 10.1016/0022-247X(80)90282-6. [32] X. Wang, A Kato type theorem on zero viscosity limit of Navier-Stokes flows, Indiana Univ. Math. J., 50 (2001), 223-241.  doi: 10.1512/iumj.2001.50.2098. [33] C. Wang, Y. Wang and Z. Zhang, Zero-viscosity limit of the Navier-Stokes equations in the analytic setting, Arch. Ration. Mech. Anal., 224 (2017), 555-595.  doi: 10.1007/s00205-017-1083-6. [34] H. Wen and S. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals, Nonlinear Anal. Real World Appl., 12 (2011), 1510-1531.  doi: 10.1016/j.nonrwa.2010.10.010. [35] Z. Xin and T. Yanagisawa, Zero-viscosity limit of the linearized Navier-Stokes equations for a compressible viscous fluid in the half-plane, Comm. Pure Appl. Math., 52 (1999), 479-541.  doi: 10.1002/(SICI)1097-0312(199904)52:4<479::AID-CPA4>3.0.CO;2-1. [36] Z. Xin and L. Zhang, On the global existence of solutions to the Prandtl's system, Adv. Math., 181 (2004), 88-133.  doi: 10.1016/S0001-8708(03)00046-X.

show all references

##### References:
 [1] R. Alexander, Y.-G. Wang, C.-J. Xu and T. Yang, Well posedness of the Prandtl equation in Sobolev spaces, J. Amer. Math. Soc., 28 (2015), 745-784.  doi: 10.1090/S0894-0347-2014-00813-4. [2] C. Bardos, M. C. Lopes Filho, D. Niu, H. J. Nussenzveig Lopes and E. S. Titi, Stability of two-dimensional viscous incompressible flows under three-dimensional perturbations and inviscid symmetry breaking, SIAM J. Math. Anal., 45 (2013), 1871-1885.  doi: 10.1137/120862569. [3] H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids, Comm. Partial Differential Equations, 28 (2003), 1183-1201.  doi: 10.1081/PDE-120021191. [4] M. Fei, T. Tao and Z. Zhang, On the zero-viscosity limit of the Navier-Stokes equations in $\mathbb{R}^3_+$ without analyticity, J. Math. Pures Appl. (9) 112 (2018), 170–229. doi: 10.1016/j.matpur.2017.09.007. [5] D. Gérard-Varet and E. Dormy, On the ill-posedness of the Prandtl equations, J. Amer. Math. Soc., 23 (2010), 591-609.  doi: 10.1090/S0894-0347-09-00652-3. [6] D. Gérard-Varet and Y. Maekawa, Sobolev stability of Prandtl expansions for the steady Navier-Stokes equations, Arch. Ration. Mech. Anal., 233 (2019), 1319-1382.  doi: 10.1007/s00205-019-01380-x. [7] G.-M. Gie, J. P. Kelliher, M. C. Lopes Filho, A. L. Mazzucato and H. J. Nussenzveig Lopes, The vanishing viscosity limit for some symmetric flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2019), 1237-1280.  doi: 10.1016/j.anihpc.2018.11.006. [8] E. Grenier, On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math., 53 (2000), 1067-1091.  doi: 10.1002/1097-0312(200009)53:9<1067::AID-CPA1>3.0.CO;2-Q. [9] Y. Guo and S. Iyer, Validity of steady Prandtl layer expansions, preprint, arXiv: 1805.05891. [10] Y. Guo and T. Nguyen, Prandtl boundary layer expansions of steady Navier-Stokes flows over a moving plate, Ann. PDE, 3 (2017), 58pp. doi: 10.1007/s40818-016-0020-6. [11] D. Han, A. L. Mazzucato, D. Niu and X. Wang, Boundary layer for a class of nonlinear pipe flow, J. Differential Equations, 252 (2012), 6387-6413.  doi: 10.1016/j.jde.2012.02.012. [12] S. Iyer, Steady Prandtl boundary layer expansions over a rotating disk, Arch. Ration. Mech. Anal., 224 (2017), 421-469.  doi: 10.1007/s00205-017-1080-9. [13] J. Kim, Weak solutions of an initial boundary value problem for an incompressible viscous fluid with nonnegative density, SIAM J. Math. Anal., 18 (1987), 89-96.  doi: 10.1137/0518007. [14] C. Liu, Y. Wang and T. Yang, A well-posedness theory for the Prandtl equations in three space variables, Adv. Math., 308 (2017), 1074-1126.  doi: 10.1016/j.aim.2016.12.025. [15] C. Liu, Y. Wang and T. Yang, On the ill-posedness of the Prandtl equations in three-dimensional space, Arch. Ration. Mech. Anal., 220 (2016), 83-108.  doi: 10.1007/s00205-015-0927-1. [16] C. Liu, F. Xie and T. Yang, MHD boundary layers theory in Sobolev spaces without monotonicity. I: Well-posedness theory, Comm. Pure Appl. Math., 72 (2019), 63-121.  doi: 10.1002/cpa.21763. [17] C. Liu, F. Xie and T. Yang, Justification of Prandtl ansatz for MHD boundary layer, SIAM J. Math. Anal., 51 (2019), 2748-2791.  doi: 10.1137/18M1219618. [18] Y. Maekawa, On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half-plane, Comm. Pure Appl. Math., 67 (2014), 1045-1128.  doi: 10.1002/cpa.21516. [19] N. Masmoudi and T. K. Wong, Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods, Comm. Pure Appl. Math., 68 (2015), 1683-1741.  doi: 10.1002/cpa.21595. [20] J. E. Marsden, Well-posedness of the equations of a non-homogeneous perfect fluid, Comm. Partial Differential Equations, 1 (1976), 215-230.  doi: 10.1080/03605307608820010. [21] A. Mazzucato, D. Niu and X. Wang, Boundary layer associated with a class of 3D nonlinear plane parallel channel fows, Indiana Univ. Math. J., 60 (2011), 1113-1136.  doi: 10.1512/iumj.2011.60.4479. [22] O. A. Oleinik, On the system of Prandtl equations in boundary-layer theory, Dokl. Akad. Nauk SSR, 150 (1963), 28-31. [23] O. A. Oleinik and V. N. Samokhin, Mathematical Models in Boundary Layer Theory, Applied Mathematics and Mathematical Computation, 15, Chapman & Hall/CRC, Boca Raton, FL, 1999. [24] L. Prandtl, Über flüssigkeitsbewegungen bei sehr kleiner reibung, Verhaldlg III Int. Math. Kong, (1905), 484–491. [25] M. Sammartino and R. E. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space, I. Existence for Euler and Prandtl equations, Comm. Math. Phys., 192 (1998), 433-461.  doi: 10.1007/s002200050304. [26] M. Sammartino and R. E. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half space, Ⅱ. Construction of the Navier-Stokes solution, Comm. Math. Phys., 192 (1998), 463-491.  doi: 10.1007/s002200050305. [27] J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of viscosity, density and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117.  doi: 10.1137/0521061. [28] R. Temam and X. Wang, Asymptotic analysis of Oseen type equations in a channel at small viscosity, Indiana Univ. Math. J., 45 (1996), 863-916.  doi: 10.1512/iumj.1996.45.1290. [29] R. Temam and X. Wang, Boundary layers associated with incompressible Navier-Stokes equations: The noncharacteristic boundary case, J. Differential Equations, 179 (2002), 647-686.  doi: 10.1006/jdeq.2001.4038. [30] H. B. da Veiga and A. Valli, On the Euler equations for nonhomogeneous fluids. (Ⅰ), Rend. Sem. Mat. Univ. Padova, 63 (1980), 151-168. [31] H. B. da Veiga and A. Valli, On the Euler equations for nonhomogeneous fluids (Ⅱ), J. Math. Anal. Appl., 73 (1980), 338-350.  doi: 10.1016/0022-247X(80)90282-6. [32] X. Wang, A Kato type theorem on zero viscosity limit of Navier-Stokes flows, Indiana Univ. Math. J., 50 (2001), 223-241.  doi: 10.1512/iumj.2001.50.2098. [33] C. Wang, Y. Wang and Z. Zhang, Zero-viscosity limit of the Navier-Stokes equations in the analytic setting, Arch. Ration. Mech. Anal., 224 (2017), 555-595.  doi: 10.1007/s00205-017-1083-6. [34] H. Wen and S. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals, Nonlinear Anal. Real World Appl., 12 (2011), 1510-1531.  doi: 10.1016/j.nonrwa.2010.10.010. [35] Z. Xin and T. Yanagisawa, Zero-viscosity limit of the linearized Navier-Stokes equations for a compressible viscous fluid in the half-plane, Comm. Pure Appl. Math., 52 (1999), 479-541.  doi: 10.1002/(SICI)1097-0312(199904)52:4<479::AID-CPA4>3.0.CO;2-1. [36] Z. Xin and L. Zhang, On the global existence of solutions to the Prandtl's system, Adv. Math., 181 (2004), 88-133.  doi: 10.1016/S0001-8708(03)00046-X.
The plane parallel channel flow in $Q = [0, L]^2 \times [0, 1]$
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