Abstract
For the two-dimensional Magnetohydrodynamics (MHD) boundary layer system, it has been shown that the non-degenerate tangential magnetic field leads to the well-posedness in Sobolev spaces and high Reynolds number limits without any monotonicity condition on the velocity field in our previous works. This paper aims to show that sufficient degeneracy in the tangential magnetic field at a non-degenerate critical point of the tangential velocity field of shear flow indeed yields instability as for the classical Prandtl equations without magnetic field studied by Gérard-Varet and Dormy (2010). This partially shows the necessity of the non-degeneracy in the tangential magnetic field for the stability of the boundary layer of MHD in 2D at least in Sobolev spaces.
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Alexander R, Wang Y-G, Xu C-J, et al. Well-posedness of the Prandtl equation in Sobolev spaces. J Amer Math Soc, 2015, 28: 745–784
E W-N, Engquist B. Blowup of solutions of the unsteady Prandtl’s equation. Comm Pure Appl Math, 1997, 50: 1287–1293
Gérard-Varet D, Dormy E. On the ill-posedness of the Prandtl equations. J Amer Math Soc, 2010, 23: 591–609
Gérard-Varet D, Masmoudi N. Well-posedness for the Prandtl system without analyticity or monotonicity. Ann Sci Éc Norm Supér (4), 2015, 48: 1273–1325
Gérard-Varet D, Nguyen T. Remarks on the ill-posedness of the Prandtl equation. Asymptot Anal, 2012, 77: 71–88
Gérard-Varet D, Prestipino M. Formal derivation and stability analysis of boundary layer models in MHD. Z Angew Math Phys, 2017, 68: 76
Grenier E. On the nonlinear instability of Euler and Prandtl equations. Comm Pure Appl Math, 2000, 53: 1067–1091
Guo Y, Nguyen T. A note on the Prandtl boundary layers. Comm Pure Appl Math, 2011, 64: 1416–1438
Li W-X, Yang T. Well-posedness in Gevrey function space for the Prandtl equations with non-degenerate critical points. Ar**v:1609.08430, 2016
Liu C-J, Wang Y-G, Yang T. Global existence of weak solutions to the three-dimensional Prandtl equations with a special structure. Discrete Contin Dyn Syst Ser S, 2016, 9: 2011–2029
Liu C-J, Wang Y-G, Yang T. On the ill-posedness of the Prandtl equations in three space dimensions. Arch Ration Mech Anal, 2016, 220: 83–108
Liu C-J, Wang Y-G, Yang T. A well-posedness theory for the Prandtl equations in three space variables. Adv Math, 2017, 308: 1074–1126
Liu C-J, **e F, Yang T. MHD boundary layers in Sobolev spaces without monotonicity, II: Convergence theory. Ar**v:1704.00523, 2017
Liu C-J, **e F, Yang T. MHD boundary layers theory in Sobolev spaces without monotonicity, I: Well-posedness theory. Comm Pure Appl Math, 2018, in press
Masmoudi N, Wong T-K. Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods. Comm Pure Appl Math, 2015, 68: 1683–1741
Oleinik O A. The Prandtl system of equations in boundary layer theory. Soviet Math Dokl, 1963, 4: 583–586
Oleinik O A, Samokhin V N. Mathematical Models in Boundary Layers Theory. Boca Raton: Chapman and Hall/CRC, 1999
Prandtl L. Über üssigkeits-bewegung bei sehr kleiner reibung. In: Verhandlungen des III. Heidelberg-Teubner-Leipzig: Internationlen Mathematiker Kongresses, 1904, 484–491
Sammartino M, Caisch R-E. 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, 1998, 192: 433–461
Sammartino M, Caisch R-E. 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, 1998, 192: 463–491
Wang Y-G, **e F, Yang T. Local well-posedness of Prandtl equations for compressible flow in two space variables. SIAM J Math Anal, 2015, 47: 321–346
**n Z-P, Zhang L. On the global existence of solutions to the Prandtl system. Adv Math, 2004, 181: 88–133
Zhang P, Zhang Z-F. Long time well-posedness of Prandtl system with small and analytic initial data. J Funct Anal, 2016, 270: 2591–2615
Acknowledgements
Cheng-Jie Liu’s research was supported by National Natural Science Foundation of China (Grant No. 11743009), Shanghai Sailing Program (Grant No. 18YF1411700) and Shanghai Jiao Tong University (Grant No. WF220441906). Feng **e’s research was supported by National Natural Science Foundation of China (Grant No.11571231). Tong Yang’s research was supported by the General Research Fund of Hong Kong, City University of Hong Kong (Grant No. 103713).
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Liu, CJ., **e, F. & Yang, T. A note on the ill-posedness of shear flow for the MHD boundary layer equations. Sci. China Math. 61, 2065–2078 (2018). https://doi.org/10.1007/s11425-017-9306-0
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DOI: https://doi.org/10.1007/s11425-017-9306-0