doi: 10.3934/era.2021082
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Well-posedness in Sobolev spaces of the two-dimensional MHD boundary layer equations without viscosity

1. 

School of Mathematics and Statistics, and Computational Science Hubei Key Laboratory, Wuhan University, 430072 Wuhan, China

2. 

School of Mathematics and Statistics, Wuhan University, 430072 Wuhan, China

* Corresponding author: Wei-Xi Li

Received  June 2021 Early access October 2021

Fund Project: The work was supported by NSF of China(Nos. 11961160716, 11871054, 11771342), the Natural Science Foundation of Hubei Province (No. 2019CFA007) and the Fundamental Research Funds for the Central Universities(No. 2042020kf0210)

We consider the two-dimensional MHD Boundary layer system without hydrodynamic viscosity, and establish the existence and uniqueness of solutions in Sobolev spaces under the assumption that the tangential component of magnetic fields dominates. This gives a complement to the previous works of Liu-Xie-Yang [Comm. Pure Appl. Math. 72 (2019)] and Liu-Wang-Xie-Yang [J. Funct. Anal. 279 (2020)], where the well-posedness theory was established for the MHD boundary layer systems with both viscosity and resistivity and with viscosity only, respectively. We use the pseudo-differential calculation, to overcome a new difficulty arising from the treatment of boundary integrals due to the absence of the diffusion property for the velocity.

Citation: Wei-Xi Li, Rui Xu. Well-posedness in Sobolev spaces of the two-dimensional MHD boundary layer equations without viscosity. Electronic Research Archive, doi: 10.3934/era.2021082
References:
[1]

R. AlexandreY.-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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H. Dietert and D. Gérard-Varet, Well-posedness of the Prandtl equations without any structural assumption, Ann. PDE, 5 (2019), Paper No. 8, 51 pp. doi: 10.1007/s40818-019-0063-6.  Google Scholar

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D. Gérard-Varet and N. Masmoudi, Well-posedness for the Prandtl system without analyticity or monotonicity, Ann. Sci. Éc. Norm. Supér. (4), 48 (2015), 1273–1325. doi: 10.24033/asens.2270.  Google Scholar

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D. Gérard-Varet and M. Prestipino, Formal derivation and stability analysis of boundary layer models in MHD, Z. Angew. Math. Phys., 68 (2017), Paper No. 76, 16 pp. doi: 10.1007/s00033-017-0820-x.  Google Scholar

[9]

Y. Guo and T. Nguyen, A note on Prandtl boundary layers, Comm. Pure Appl. Math., 64 (2011), 1416-1438.  doi: 10.1002/cpa.20377.  Google Scholar

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M. Ignatova and V. Vicol, Almost global existence for the Prandtl boundary layer equations, Arch. Ration. Mech. Anal., 220 (2016), 809-848.  doi: 10.1007/s00205-015-0942-2.  Google Scholar

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W.-X. Li, N. Masmoudi and T. Yang, Well-posedness in Gevrey function space for 3D Prandtl equations without structural assumption, Comm. Pure Appl. Math., (2021). doi: 10.1002/cpa.21989.  Google Scholar

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W.-X. LiD. Wu and C.-J. Xu, Gevrey class smoothing effect for the Prandtl equation, SIAM J. Math. Anal., 48 (2016), 1672-1726.  doi: 10.1137/15M1020368.  Google Scholar

[14]

W.-X. Li and T. Yang, Well-posedness in Gevrey function spaces for the Prandtl equations with non-degenerate critical points, J. Eur. Math. Soc. (JEMS), 22 (2020), 717-775.  doi: 10.4171/jems/931.  Google Scholar

[15]

W.-X. Li and T. Yang, Well-posedness of the MHD boundary layer system in Gevrey function space without structural assumption, SIAM J. Math. Anal., 53 (2021), 3236-3264.  doi: 10.1137/20M1367027.  Google Scholar

[16]

C.-J. Liu, D. Wang, F. Xie and T. Yang, Magnetic effects on the solvability of 2D MHD boundary layer equations without resistivity in Sobolev spaces, J. Funct. Anal., 279 (2020), 108637, 45 pp. doi: 10.1016/j.jfa.2020.108637.  Google Scholar

[17]

C.-J. 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]

C.-J. 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

[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]

O. A. Oleinik and V. N. Samokhin, Mathematical Models in Boundary Layer Theory, vol. 15 of Applied Mathematics and Mathematical Computation, Chapman & Hall/CRC, Boca Raton, FL, 1999.  Google Scholar

[21]

M. Ruzhansky and V. Turunen, On the {F}ourier analysis of operators on the torus, in Modern Trends in Pseudo-Differential Operators, vol. 172 of Oper. Theory Adv. Appl., Birkhäuser, Basel, (2007), 87–105. doi: 10.1007/978-3-7643-8116-5_5.  Google Scholar

[22]

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

[23]

C.-J. Xu and X. Zhang, Long time well-posedness of Prandtl equations in Sobolev space, J. Differential Equations, 263 (2017), 8749-8803.  doi: 10.1016/j.jde.2017.08.046.  Google Scholar

[24]

P. Zhang and Z. Zhang, Long time well-posedness of Prandtl system with small and analytic initial data, J. Funct. Anal., 270 (2016), 2591-2615.  doi: 10.1016/j.jfa.2016.01.004.  Google Scholar

show all references

References:
[1]

R. AlexandreY.-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]

D. ChenY. Wang and Z. Zhang, Well-posedness of the linearized Prandtl equation around a non-monotonic shear flow, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 1119-1142.  doi: 10.1016/j.anihpc.2017.11.001.  Google Scholar

[3]

H. ChenW.-X. Li and C.-J. Xu, Gevrey hypoellipticity for a class of kinetic equations, Comm. Partial Differential Equations, 36 (2011), 693-728.  doi: 10.1080/03605302.2010.507689.  Google Scholar

[4]

H. Dietert and D. Gérard-Varet, Well-posedness of the Prandtl equations without any structural assumption, Ann. PDE, 5 (2019), Paper No. 8, 51 pp. doi: 10.1007/s40818-019-0063-6.  Google Scholar

[5]

W. E and B. Engquist, Blowup of solutions of the unsteady Prandtl's equation, Comm. Pure Appl. Math., 50 (1997), 1287-1293.  doi: 10.1002/(SICI)1097-0312(199712)50:12<1287::AID-CPA4>3.0.CO;2-4.  Google Scholar

[6]

D. Gérard-Varet and E. Dormy, On the ill-posedness of the Prandtl equation, J. Amer. Math. Soc., 23 (2010), 591-609.  doi: 10.1090/S0894-0347-09-00652-3.  Google Scholar

[7]

D. Gérard-Varet and N. Masmoudi, Well-posedness for the Prandtl system without analyticity or monotonicity, Ann. Sci. Éc. Norm. Supér. (4), 48 (2015), 1273–1325. doi: 10.24033/asens.2270.  Google Scholar

[8]

D. Gérard-Varet and M. Prestipino, Formal derivation and stability analysis of boundary layer models in MHD, Z. Angew. Math. Phys., 68 (2017), Paper No. 76, 16 pp. doi: 10.1007/s00033-017-0820-x.  Google Scholar

[9]

Y. Guo and T. Nguyen, A note on Prandtl boundary layers, Comm. Pure Appl. Math., 64 (2011), 1416-1438.  doi: 10.1002/cpa.20377.  Google Scholar

[10]

M. Ignatova and V. Vicol, Almost global existence for the Prandtl boundary layer equations, Arch. Ration. Mech. Anal., 220 (2016), 809-848.  doi: 10.1007/s00205-015-0942-2.  Google Scholar

[11]

I. KukavicaN. MasmoudiV. Vicol and T. K. Wong, On the local well-posedness of the Prandtl and hydrostatic Euler equations with multiple monotonicity regions, SIAM J. Math. Anal., 46 (2014), 3865-3890.  doi: 10.1137/140956440.  Google Scholar

[12]

W.-X. Li, N. Masmoudi and T. Yang, Well-posedness in Gevrey function space for 3D Prandtl equations without structural assumption, Comm. Pure Appl. Math., (2021). doi: 10.1002/cpa.21989.  Google Scholar

[13]

W.-X. LiD. Wu and C.-J. Xu, Gevrey class smoothing effect for the Prandtl equation, SIAM J. Math. Anal., 48 (2016), 1672-1726.  doi: 10.1137/15M1020368.  Google Scholar

[14]

W.-X. Li and T. Yang, Well-posedness in Gevrey function spaces for the Prandtl equations with non-degenerate critical points, J. Eur. Math. Soc. (JEMS), 22 (2020), 717-775.  doi: 10.4171/jems/931.  Google Scholar

[15]

W.-X. Li and T. Yang, Well-posedness of the MHD boundary layer system in Gevrey function space without structural assumption, SIAM J. Math. Anal., 53 (2021), 3236-3264.  doi: 10.1137/20M1367027.  Google Scholar

[16]

C.-J. Liu, D. Wang, F. Xie and T. Yang, Magnetic effects on the solvability of 2D MHD boundary layer equations without resistivity in Sobolev spaces, J. Funct. Anal., 279 (2020), 108637, 45 pp. doi: 10.1016/j.jfa.2020.108637.  Google Scholar

[17]

C.-J. 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]

C.-J. 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

[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]

O. A. Oleinik and V. N. Samokhin, Mathematical Models in Boundary Layer Theory, vol. 15 of Applied Mathematics and Mathematical Computation, Chapman & Hall/CRC, Boca Raton, FL, 1999.  Google Scholar

[21]

M. Ruzhansky and V. Turunen, On the {F}ourier analysis of operators on the torus, in Modern Trends in Pseudo-Differential Operators, vol. 172 of Oper. Theory Adv. Appl., Birkhäuser, Basel, (2007), 87–105. doi: 10.1007/978-3-7643-8116-5_5.  Google Scholar

[22]

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

[23]

C.-J. Xu and X. Zhang, Long time well-posedness of Prandtl equations in Sobolev space, J. Differential Equations, 263 (2017), 8749-8803.  doi: 10.1016/j.jde.2017.08.046.  Google Scholar

[24]

P. Zhang and Z. Zhang, Long time well-posedness of Prandtl system with small and analytic initial data, J. Funct. Anal., 270 (2016), 2591-2615.  doi: 10.1016/j.jfa.2016.01.004.  Google Scholar

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