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 & Continuous Dynamical Systems - A, 2020, 40 (8) : 4579-4596. doi: 10.3934/dcds.2020193
References:
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R. AlexanderY.-G. WangC.-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.  Google Scholar

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C. BardosM. C. Lopes FilhoD. NiuH. 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.  Google Scholar

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[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.  Google Scholar

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Y. Guo and S. Iyer, Validity of steady Prandtl layer expansions, preprint, arXiv: 1805.05891. Google Scholar

[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.  Google Scholar

[11]

D. HanA. L. MazzucatoD. 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[14]

C. LiuY. 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.  Google Scholar

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C. LiuF. Xie and T. Yang, Justification of Prandtl ansatz for MHD boundary layer, SIAM J. Math. Anal., 51 (2019), 2748-2791.  doi: 10.1137/18M1219618.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[21]

A. MazzucatoD. 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.  Google Scholar

[22]

O. A. Oleinik, On the system of Prandtl equations in boundary-layer theory, Dokl. Akad. Nauk SSR, 150 (1963), 28-31.   Google Scholar

[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.  Google Scholar

[24]

L. Prandtl, Über flüssigkeitsbewegungen bei sehr kleiner reibung, Verhaldlg III Int. Math. Kong, (1905), 484–491. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

C. WangY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

R. AlexanderY.-G. WangC.-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.  Google Scholar

[2]

C. BardosM. C. Lopes FilhoD. NiuH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

G.-M. GieJ. P. KelliherM. C. Lopes FilhoA. 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.  Google Scholar

[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.  Google Scholar

[9]

Y. Guo and S. Iyer, Validity of steady Prandtl layer expansions, preprint, arXiv: 1805.05891. Google Scholar

[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.  Google Scholar

[11]

D. HanA. L. MazzucatoD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

C. LiuY. 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.  Google Scholar

[15]

C. LiuY. 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.  Google Scholar

[16]

C. LiuF. 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.  Google Scholar

[17]

C. LiuF. Xie and T. Yang, Justification of Prandtl ansatz for MHD boundary layer, SIAM J. Math. Anal., 51 (2019), 2748-2791.  doi: 10.1137/18M1219618.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

A. MazzucatoD. 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.  Google Scholar

[22]

O. A. Oleinik, On the system of Prandtl equations in boundary-layer theory, Dokl. Akad. Nauk SSR, 150 (1963), 28-31.   Google Scholar

[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.  Google Scholar

[24]

L. Prandtl, Über flüssigkeitsbewegungen bei sehr kleiner reibung, Verhaldlg III Int. Math. Kong, (1905), 484–491. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

C. WangY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  The plane parallel channel flow in $ Q = [0, L]^2 \times [0, 1] $
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