Local existence of solutions to the 2D MHD boundary layer equations without monotonicity in Sobolev space

Xiaolei Dong; Xiaolei Dong

doi:10.3934/math.2024256

AIMS Mathematics

2024, Volume 9, Issue 3: 5294-5329. doi: 10.3934/math.2024256

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Research article

Local existence of solutions to the 2D MHD boundary layer equations without monotonicity in Sobolev space

Xiaolei Dong ^,

School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou 466001, China

Received: 15 November 2023 Revised: 19 January 2024 Accepted: 19 January 2024 Published: 26 January 2024
MSC : 35M33, 35Q35, 76D03, 76D10, 76W05

In this work, we investigated the local existence of the solutions to the 2D magnetohy-drodynamic (MHD) boundary layer equations on the half plane by energy methods in weighted Sobolev space. Compared to the existence of solutions to the classical Prandtl equations where the monotonicity condition of the tangential velocity plays an important role, we used the initial tangential magnetic field with a lower bound $ \delta > 0 $ instead of the monotonicity condition of the tangential velocity.
- MHD boundary layer equations,
- the existence of solutions,
- the energy method,
- the weighted sobolev space
Citation: Xiaolei Dong. Local existence of solutions to the 2D MHD boundary layer equations without monotonicity in Sobolev space[J]. AIMS Mathematics, 2024, 9(3): 5294-5329. doi: 10.3934/math.2024256

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Abstract

In this work, we investigated the local existence of the solutions to the 2D magnetohy-drodynamic (MHD) boundary layer equations on the half plane by energy methods in weighted Sobolev space. Compared to the existence of solutions to the classical Prandtl equations where the monotonicity condition of the tangential velocity plays an important role, we used the initial tangential magnetic field with a lower bound $ \delta > 0 $ instead of the monotonicity condition of the tangential velocity.

References

[1]	R. A. Adams, J. J. F. Fournier, Sobolev Spaces, New York-London: Academic Press, 2003.
[2]	R. Alexandre, Y. G. Wang, C. J. Xu, T. Yang, Well-posedness of the Prandtl equation in Sobolev spaces, J. Am. Math. Soc., 28 (2015), 745–784. http://doi.org/10.1090/S0894-0347-2014-00813-4 doi: 10.1090/S0894-0347-2014-00813-4
[3]	T. G. Cowling, Magnetohydrodynamics, New York: Interscience Publishers Inc., 1957.
[4]	D. X. Chen, X. L. Li, Long time well-posedness of two dimensional Magnetohydrodynamic boundary layer equation without resistivity, Math. Method. Appl. Sci., 46 (2023), 10186–10202. http://doi.org/10.1002/mma.9110 doi: 10.1002/mma.9110
[5]	P. A. Davidson, An introduction to magnetohydrodynamics, Cambridge: Cambridge University Press, 2001. https://doi.org/10.1017/CBO9780511626333
[6]	P. G. Drasin, Stability of parallel flow in a parallel magnetic field at small magnetic Reynolds number, J. Fluid Mech., 8 (1960), 130–142. http://doi.org/10.1017/S0022112060000475 doi: 10.1017/S0022112060000475
[7]	J. C. Gao, D. E. Huang, Z. A. Yao, Boundary layer problems for the two-dimensional inhomogeneous incompressible magnetohydrodynamics equations, Calc. Var., 60 (2021), 67. http://doi.org/10.1007/s00526-021-01958-y doi: 10.1007/s00526-021-01958-y
[8]	D. G$\acute{e}$rard-Varet, E. Dormy, On the ill-posedness of the Prandtl equations, J. Am. Math. Soc., 23 (2010), 591–609.
[9]	D. G$\acute{e}$rard-Varet, M. Prestipino, Formal derivation and stability analysis of boundary layer models in MHD, Z. Angew. Math. Phys., 68 (2016), 76. http://doi.org/10.1007/s00033-017-0820-x doi: 10.1007/s00033-017-0820-x
[10]	Y. T. Huang, C. J. Liu, T. Yang, Local-in-time well-posedness for compressible MHD boundary layer, J. Differ. Equations, 266 (2019), 2978–3013. http://doi.org/10.1016/j.jde.2018.08.052 doi: 10.1016/j.jde.2018.08.052
[11]	W. X. Li, R. Xu, Well-posedness in sobolev spaces of the two-dimensional MHD boundary layer equations without viscosity, Electron. Res. Arch., 29 (2021), 4243–4255. http://doi.org/10.3934/era.2021082 doi: 10.3934/era.2021082
[12]	C. J. Liu, D. H. Wang, F. Xie, T. Yang, Magnetic effects on the solvability of 2D MHD boundary layer equations without resistivity in Sobolev spaces, J. Funct. Anal., 279 (2020), 108637. http://doi.org/10.1016/j.jfa.2020.108637 doi: 10.1016/j.jfa.2020.108637
[13]	C. J. Liu, F. Xie, T. Yang, A note on the ill-posedness of shear flow for the MHD boundary layer equations, Sci. China Math., 61 (2018), 2065–2078. http://doi.org/10.1007/s11425-017-9306-0 doi: 10.1007/s11425-017-9306-0
[14]	C. J. Liu, F. Xie, T. Yang, MHD Boundary layers theory in Sobolev spaces without monotonicity Ⅰ: well-posedness theory, Commun. Pur. Appl. Math., 72 (2019), 63–121. http://doi.org/10.1002/cpa.21763 doi: 10.1002/cpa.21763
[15]	C. J. Liu, F. Xie, T. Yang, MHD boundary layers in Sobolev spaces without monotonicity Ⅱ: convergence theory, 2017, arXiv: 1704.00523.
[16]	N. Liu, P. Zhang, Global small analytic solutions of MHD boundary layer equations, J. Differ. Equations, 281 (2021), 199–257. http://doi.org/10.1016/j.jde.2021.02.003 doi: 10.1016/j.jde.2021.02.003
[17]	N. Masmoudi, T. K. Wong, Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods, Commun. Pur. Appl. Math., 68 (2015), 1683–1741. http://doi.org/10.1002/cpa.21595 doi: 10.1002/cpa.21595
[18]	D. S. Mitrinovi$\acute{c}$, J. E. P$\check{e}$cari$\acute{c}$, A. M. Fink, Inequalities involving functions and their integrals and derivatives, Dordrecht: Springer, 1991. https://doi.org/10.1007/978-94-011-3562-7
[19]	O. A. Oleinik, The Prandtl system of equations in boundary layer theory, Dokl. Acad. Nauk. SSSR, 150 (1963), 28–32.
[20]	O. A. Oleinik, V. N. Samokhin, Mathematical models in boundary layer theory, New York: Routledge, 1999. https://doi.org/10.1201/9780203749364
[21]	L. Prandtl, $\ddot{u}$ber Fl$\ddot{u}$ssigkeitsbewegung bei sehr kleiner reibung, The International Mathematical Congress, Heidelberg, Germany, 1904,484–491.
[22]	V. J. Rossow, Boundary layer stability diagrams for electrically conducting fluids in the presence of a magnetic field, NACA Technical Note, 1958 (1958), 4282.
[23]	M. Sermange, R. Temam, Some mathematical questions related to the MHD equations, Commun. Pur. Appl. Math., 36 (1983), 635–664. http://doi.org/10.1002/cpa.3160360506 doi: 10.1002/cpa.3160360506
[24]	H. Triebel, Theory of function spaces, Basel: Birkhäuser, 1983. https://doi.org/10.1007/978-3-0346-0416-1
[25]	S. Wang, N. Wang, Boundary layer problem of MHD system with non-characteristic perfect conducting wall, Appl. Anal., 98 (2017), 1–20. http://doi.org/10.1080/00036811.2017.1395867 doi: 10.1080/00036811.2017.1395867
[26]	S. Wang, Z. P. Xin, Boundary layer problems in the viscosity diffusion vanishing limits for the incompressible MHD systems, Scientia Sinica Mathematica, 47 (2017), 1303–1326. http://doi.org/10.1360/N012016-00211 doi: 10.1360/N012016-00211
[27]	S. Wang, B. Y. Wang, C. D. Liu, N. Wang, Boundary layer problem and zero viscosity-diffusion limit of the incompressible magnetohydrodynamic system with no-slip boundary conditions, J. Differ. Equations, 263 (2017), 4723–4749. http://doi.org/10.1016/j.jde.2017.05.025 doi: 10.1016/j.jde.2017.05.025
[28]	F. Xie, T. Yang, Global-in-time stability of 2D MHD boundary layer in the Prandtl-Hartmann regime, SIAM J. Math. Anal., 50 (2018), 5749–5760. http://doi.org/10.1137/18M1174969 doi: 10.1137/18M1174969
[29]	F. Xie, T. Yang, Lifespan of solutions to MHD Boundary layer equations with analytic perturbation of general shear flow, Acta Math. Appl. Sin. Engl. Ser., 35 (2019), 209–229. http://doi.org/10.1007/s10255-019-0805-y doi: 10.1007/s10255-019-0805-y
[30]	Z. P. Xin, L. Q. Zhang, On the global existence of solutions to the Prandtl's system, Adv. Math., 181 (2004), 88–133. https://doi.org/10.1016/S0001-8708(03)00046-X doi: 10.1016/S0001-8708(03)00046-X
[31]	C. J. Xu, X. Zhang, Long time well-posedness of Prandtl equations in Sobolev space, J. Differ. Equations, 263 (2017), 8749–8803. http://doi.org/10.1016/j.jde.2017.08.046 doi: 10.1016/j.jde.2017.08.046

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