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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 |
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.
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. |

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