Article
| Title: | Boundary layer separation and local behavior of the steady Prandtl equation with magnetic damping (English) |
| Author: | Chen, Yanjun |
| Author: | Zhang, Xinhan |
| Author: | Luo, Hong |
| Language: | English |
| Journal: | Applications of Mathematics |
| ISSN: | 0862-7940 (print) |
| ISSN: | 1572-9109 (online) |
| Volume: | 71 |
| Issue: | 3 |
| Year: | 2026 |
| Pages: | 327-351 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | We prove the boundary layer separation of the two-dimensional steady Prandtl equation with magnetic damping under an appropriate adverse pressure gradient. We also obtain the separation rate of the solution and the local behavior of the solution near the separation point. Our results show that magnetic damping exerts a suppressive effect on boundary layer separation. (English) |
| Keyword: | boundary layer separation |
| Keyword: | separation rate |
| Keyword: | magnetic damping |
| MSC: | 76D10 |
| MSC: | 76W05 |
| DOI: | 10.21136/AM.2026.0298-25 |
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| Date available: | 2026-07-02T06:51:11Z |
| Last updated: | 2026-08-19 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153671 |
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| Reference: | [1] Bhattacharyya, K., Pop, I.: MHD boundary layer flow due to an exponentially shrinking sheet.Magnetohydrodynamics 47 (2011), 337-344. 10.22364/mhd.47.4.2 |
| Reference: | [2] Brown, S. N., Stewartson, K.: Laminar separation.Annual Review of Fluid Mechanics. Vol. 1 Annual Reviews, Palo Alto (1969), 45-72. Zbl 0207.25101, 10.1146/annurev.fl.01.010169.000401 |
| Reference: | [3] Buckmaster, J.: Separation and magnetohydrodynamics.J. Fluid Mech. 38 (1969), 481-498. Zbl 0187.25302, 10.1017/S0022112069000292 |
| Reference: | [4] Bulatova, R. R., Chechkin, G. A., Chechkina, T. P., Samokhin, V. N.: On the influence of a magnetic field on the separation of the boundary layer of a non-Newtonian MHD medium.C. R. Mecanique 346 (2018), 807-814. 10.1016/j.crme.2018.06.010 |
| Reference: | [5] Bulatova, P. P., Samokhin, V. N., Chechkin, G. A.: Equations of magnetohydrodynamic boundary layer for a modified incompressible viscous medium: Boundary layer separation.J. Math. Sci., New York 232 (2018), 299-321. Zbl 1397.76159, MR 3813482, 10.1007/s10958-018-3874-1 |
| Reference: | [6] Chan, C., Czubak, M., Yoneda, T.: An ODE for boundary layer separation on a sphere and a hyperbolic space.Physica D 282 (2014), 34-38. Zbl 1349.76076, MR 3217422, 10.1016/j.physd.2014.05.004 |
| Reference: | [7] Chen, D., Ren, S., Wang, Y., Zhang, Z.: Global well-posedness of the 2-D magnetic Prandtl model in the Prandtl-Hartmann regime.Asymptotic Anal. 120 (2020), 373-393. Zbl 1472.35288, MR 4169212, 10.3233/ASY-191593 |
| Reference: | [8] Dalibard, A.-L., Masmoudi, N.: Separation for the stationary Prandtl equation.Publ. Math., Inst. Hautes Étud. Sci. 130 (2019), 187-297. Zbl 1427.35201, MR 4028516, 10.1007/s10240-019-00110-z |
| Reference: | [9] E, W.: Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation.Acta Math. Sin., Engl. Ser. 16 (2000), 207-218. Zbl 0961.35101, MR 1778702, 10.1007/s101140000034 |
| Reference: | [10] Gao, J., Li, M., Yao, Z.: Higher regularity and asymptotic behavior of 2D magnetic Prandtl model in the Prandtl-Hartmann regime.J. Differ. Equations 386 (2024), 294-367. Zbl 1533.35272, MR 4687368, 10.1016/j.jde.2023.12.030 |
| Reference: | [11] Gérard-Varet, D., Prestipino, M.: Formal derivation and stability analysis of boundary layer models in MHD.Z. Angew. Math. Phys. 68 (2017), Article ID 76, 16 pages. Zbl 1432.76285, MR 3657241, 10.1007/s00033-017-0820-x |
| Reference: | [12] Ghil, M., Liu, J.-G., Wang, C., Wang, S.: Boundary-layer separation and adverse pressure gradient for 2-D viscous incompressible flow.Physica D 197 (2004), 149-173. Zbl 1054.76025, MR 2092098, 10.1016/j.physd.2004.06.012 |
| Reference: | [13] Goldstein, S.: On laminar boundary-layer flow near a position of separation.Q. J. Mech. Appl. Math. 1 (1948), 43-69. Zbl 0033.31701, MR 0027197, 10.1093/qjmam/1.1.43 |
| Reference: | [14] Han, D., Mazzucato, A., Niu, D., Wang, X.: Boundary layer for a class of nonlinear pipe flow.J. Differ. Equations 252 (2012), 6387-6413. Zbl 1246.35159, MR 2911839, 10.1016/j.jde.2012.02.012 |
| Reference: | [15] Ju, Q., Luo, T., Xu, X.: Singular limits for the Navier-Stokes-Poisson equations of the viscous plasma with the strong density boundary layer.Sci. China, Math. 66 (2023), 1495-1528. Zbl 1524.76337, MR 4603435, 10.1007/s11425-022-2008-8 |
| Reference: | [16] Leibovich, S.: Magnetohydrodynamic flow at a rear stagnation point.J. Fluid Mech. 29 (1967), 401-413. Zbl 0204.28905, 10.1017/S0022112067000916 |
| Reference: | [17] Li, S., Xie, F.: Global solvability of 2D MHD boundary layer equations in analytic function spaces.J. Differ. Equations 299 (2021), 362-401. Zbl 1476.35205, MR 4293727, 10.1016/j.jde.2021.07.025 |
| Reference: | [18] Li, S., Xie, F.: Long time well-posedness of compressible magnetohydrodynamics boundary layer equations in Sobolev space.Discrete Contin. Dyn. Syst. 44 (2024), 943-969. Zbl 1546.76116, MR 4696147, 10.3934/dcds.2023133 |
| Reference: | [19] Li, W.-X., Xu, R., Yang, T.: Global well-posedness of a Prandtl model from MHD in Gevrey function spaces.Acta Math. Sci., Ser. B, Engl. Ed. 42 (2022), 2343-2366. Zbl 1513.76172, MR 4494626, 10.1007/s10473-022-0609-7 |
| Reference: | [20] Liron, N., Wilhelm, H. E.: Integration of the magnetohydrodynamic boundary-layer equations by Meksyn's method.Z. angew. Math. Mech. 54 (1974), 27-37. Zbl 0272.76061, 10.1002/zamm.19740540105 |
| Reference: | [21] Liu, C.-J., Wang, D., Xie, F., Yang, T.: Magnetic effects on the solvability of 2D MHD boundary layer equations without resistivity in Sobolev spaces.J. Funct. Anal. 279 (2020), Article ID 108637, 44 pages. Zbl 1445.76097, MR 4102162, 10.1016/j.jfa.2020.108637 |
| Reference: | [22] Liu, C.-J., Xie, F., Yang, T.: Justification of Prandtl ansatz for MHD boundary layer.SIAM J. Math. Anal. 51 (2019), 2748-2791. Zbl 1419.76555, MR 3975147, 10.1137/18M1219618 |
| Reference: | [23] Liu, C.-J., Xie, F., Yang, T.: MHD boundary layers theory in Sobolev spaces without monotonicity. I. Well-posedness theory.Commun. Pure Appl. Math. 72 (2019), 63-121. Zbl 1404.35492, MR 3882222, 10.1002/cpa.21763 |
| Reference: | [24] Liu, C.-J., Yang, T., Zhang, Z.: Analysis of the Tollmien-Schlichting wave in the Prandtl-Hartmann regime.J. Math. Pures Appl. (9) 165 (2022), 58-105. Zbl 1496.35321, MR 4470110, 10.1016/j.matpur.2022.07.004 |
| Reference: | [25] Oleinik, O. A., Samokhin, V. N.: Mathematical Models in Boundary Layer Theory.Applied Mathematics and Mathematical Computation 15. Chapman & Hall/CRC, Boca Raton (1999). Zbl 0928.76002, MR 1697762 |
| Reference: | [26] Prandtl, L.: Über Flüssigkeitsbewegung bei sehr kleiner Reibung.Verhandlungen des Dritten Internationalen Mathematiker-Kongresses in Heidelberg vom 8. bis 13. August 1904 B. G. Teubner, Leipzig (1905), 484-491 German \99999JFM99999 36.0800.02. |
| Reference: | [27] Qin, Y., Dong, X., Wang, X.: Prandtl Equations and Related Boundary Layer Equations.Springer, Singapore (2024). Zbl 1557.76005, MR 4890567, 10.1007/978-981-97-4565-4 |
| Reference: | [28] Rossow, V. J.: Boundary-Layer Stability Diagrams for Electrically Conducting Fluids in the Presence of a Magnetic Field: Technical Note 4282.National Advisory Committee for Aeronautics, Washington (1959). |
| Reference: | [29] Schlichting, H., Gersten, K.: Boundary-Layer Theory.Springer, Berlin (2017). Zbl 1358.76001, MR 3560442, 10.1007/978-3-662-52919-5 |
| Reference: | [30] Shen, W., Wang, Y., Zhang, Z.: Boundary layer separation and local behavior for the steady Prandtl equation.Adv. Math. 389 (2021), Article ID 107896, 25 pages. Zbl 1477.35178, MR 4289038, 10.1016/j.aim.2021.107896 |
| Reference: | [31] Uruba, V., Knob, M.: Dynamics of a boundary layer separation.Eng. Mech. 16 (2009), 29-38. |
| Reference: | [32] Wang, Y.-G., Xie, F., Yang, T.: Local well-posedness of Prandtl equations for compressible flow in two space variables.SIAM J. Math. Anal. 47 (2015), 321-346. Zbl 1326.35290, MR 3296610, 10.1137/140978466 |
| Reference: | [33] Wang, Y.-G., Zhu, S.-Y.: Well-posedness of thermal boundary layer equation in two-dimensional incompressible heat conducting flow with analytic datum.Math. Methods Appl. Sci. 43 (2020), 4683-4716. Zbl 1440.76022, MR 4085647, 10.1002/mma.6226 |
| Reference: | [34] Xie, F., Yang, T.: Global-in-time stability of 2D MHD boundary layer in the Prandtl-Hartmann regime.SIAM J. Math. Anal. 50 (2018), 5749-5760. Zbl 1402.76111, MR 3873032, 10.1137/18M1174969 |
| Reference: | [35] Zou, Y., Wang, J.: Boundary layer separation for the steady compressible Prandtl equation.J. Math. Anal. Appl. 527 (2023), Article ID 127379, 19 pages. Zbl 1530.35240, MR 4589207, 10.1016/j.jmaa.2023.127379 |
| Reference: | [36] Zou, Y., Xu, X., Liu, G.: Separation of the two-dimensional steady MHD boundary layer.Physica D 459 (2024), Article ID 134049, 7 pages. Zbl 1535.76031, MR 4688140, 10.1016/j.physd.2023.134049 |
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