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On a Global Weak Solution and Back Flow of the Mixed Prandtl–Hartmann Boundary Layer Problem

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Abstract

The proposal of this paper is to study the effect of magnetic field in stabilizing the hydrodynamic flow and preventing the occurrence of back flow in the Prandtl–Hartmann boundary layer. When both of the initial tangential velocity and upstream velocity are strictly increasing in the normal variable of the boundary, and the normal component of the magnetic field at the boundary is fairly strong, we obtain the global existence of a weak solution to the mixed Prandtl–Hartmann boundary layer problem, even for certain adverse pressure gradient, that may lead to the occurrence of a back-flow point for the classical Prandtl equation. This indicates that the magnetic field has certain stabilization effect on the hydrodynamic flow. On the other hand, under the monotonicity condition, we obtain that a first back-flow point, when the boundary layer evolves in time, should appear on the boundary if it occurs, notably the pressure gradient of the outer flow is not necessarily adverse. Moreover, when the adverse pressure gradient of the outer flow dominates the orthogonal magnetic field, and the initial velocity satisfies certain growth condition, we obtain the existence of a back-flow point of the Prandtl–Hartmann boundary layer.

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References

  1. Alexandre, R., Wang, Y.G., Xu, C.J., Yang, T.: Well-posedness of the Prandtl equation in Sobolev spaces. J. Am. Math. Soc. 28, 745–784 (2015)

    Article  MathSciNet  Google Scholar 

  2. Caflisch, R.E., Sammartino, M.: Existence and singularities for the Prandtl boundary layer equations. Z. Angew. Math. Mech. 80, 733–744 (2000)

    Article  MathSciNet  Google Scholar 

  3. Collot, C., Ghoul, T., Ibrahim, S. & Masmoudi, N.: On singularity formation for the two dimensional unsteady Prandtl’s system, arXiv:1808.05967v1, 2018

  4. Dalibard, A.L., Masmoudi, N.: Separation for the stationary Prandtl equation. Publ. Math. Inst. Hautes Études Sci. 130, 187–297 (2019)

    Article  MathSciNet  Google Scholar 

  5. Gérard-Varet, D., Dormy, E.: On the ill-posedness of the Prandtl equation. J. Am. Math. Soc. 23, 591–609 (2010)

    Article  MathSciNet  Google Scholar 

  6. Gérard-Varet, D., Prestipino, M.: Formal derivation and stability analysis of boundary layer models in MHD. Z. Angew. Math. Phys. 68, 76 (2017)

    Article  MathSciNet  Google Scholar 

  7. Gérard-Varet, D., Nguyen, T.T.: Remarks on the ill-posedness of the Prandtl equation. Asym. Anal. 77, 71–88 (2012)

    MathSciNet  MATH  Google Scholar 

  8. Goldstein, S.: On laminar boundary-layer flow near a position of separation. Quart. J. Mech. Appl. Math. 1, 43–69 (1948)

    Article  MathSciNet  Google Scholar 

  9. Grenier, E.: On the nonlinear instability of Euler and Prandtl equations. Commun. Pure Appl. Math. 53, 1067–1091 (2000)

    Article  MathSciNet  Google Scholar 

  10. Guo, Y., Nguyen, T.T.: A note on Prandtl boundary layers. Commun. Pure Appl. Math. 64, 1416–1438 (2011)

    Article  MathSciNet  Google Scholar 

  11. Kukavica, I., Vicol, V., Wang, F.: The van Dommelen and Shen singularity in the Prandtl equations. Adv. Math. 307, 288–311 (2017)

    Article  MathSciNet  Google Scholar 

  12. Liu, C.J., Xie, F., Yang, T.: MHD boundary layers theory in Sobolev spaces without monotonicity \(I\): well-posedness theory. Comm. Pure Appl. Math. 72, 63–121 (2019)

    Article  MathSciNet  Google Scholar 

  13. 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 9, 2011–2029 (2016)

    Article  MathSciNet  Google Scholar 

  14. Liu, C.J., Xie, F., Yang, T.: Justification of Prandtl ansatz for MHD boundary layer. SIAM J. Math. Anal. 51, 2748–2791 (2019)

    Article  MathSciNet  Google Scholar 

  15. Majda, A.J., Andrew, J., Bertozzi, A.L.: Vorticity and Incompressible Flow, vol. 27. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  16. Masmoudi, N., Wong, T.K.: On the \(H^s\) theory of hydrostatic Euler equations. Arch. Rat. Mech. Anal. 204, 231–271 (2012)

    Article  MathSciNet  Google Scholar 

  17. Masmoudi, N., Wong, T.K.: Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods. Comm. Pure Appl. Math. 68, 1683–1741 (2015)

    Article  MathSciNet  Google Scholar 

  18. Moore, F. K.: On the separation of unsteady boundary layer, Boundary-layer Research (ed. H. G. Gortler), Berlin, Germany, Springer, 1958, 296–311

  19. Oleinik, O.A.: The Prandtl system of equations in boundary layer theory. Soviet Math Dokl. 3, 583–586 (1963)

    MATH  Google Scholar 

  20. Oleinik, O.A.: On the system of equations of the theory of boundary layer. Zhurn. Vychisl. Mat. iMat. Fiz. 3, 489–507 (1963)

    MathSciNet  Google Scholar 

  21. Oleinik, O.A., Samokhin, V.N.: Mathematical Models in Boundary Layer Theory. Chapman & Hall/CRC Press, Boca Raton (1999)

    MATH  Google Scholar 

  22. Prandtl, L.: Über flüssigkeitsbewegungen bei sehr kleiner Reibung, pp. 484–494. Verh. Int. Math. Kongr, Heidelberg (1904)

    Google Scholar 

  23. Renardy, M.: Well-Posedness of the hydrostatic MHD equations. J. Math. Fluid Mech. 14, 355–361 (2012)

    Article  MathSciNet  ADS  Google Scholar 

  24. Rott, N.: Unsteady viscous flow in the vicinity of a stagnation point. Q. Appl. Math. 13, 444–451 (1956)

    Article  MathSciNet  Google Scholar 

  25. Sears, W.R., Tellionis, D.P.: Boundary-layer separation in unsteady flow. SIAM J. Appl. Math. 28, 215–235 (1975)

    Article  ADS  Google Scholar 

  26. Shen, W. M., Wang, Y. & Zhang, F. Z.: Boundary layer separation and local behavior for the steady Prandtl equation, arXiv:1904.08055, 2019

  27. Simon, J.: Compact sets in the space \(L^p(0, T;B)\). Ann. Mat. Pura Appl. 146, 65–96 (1987)

    Article  MathSciNet  Google Scholar 

  28. Stewartson, K.: On Goldstein’s theory of laminar separation. Quart. J. Mech. Appl. Math. 11, 399–410 (1958)

    Article  MathSciNet  Google Scholar 

  29. Suslov, A.I.: On the effect of injection on boundary layer separation. Prikl. Mat. Mekh. 38, 166–169 (1974)

    MathSciNet  MATH  Google Scholar 

  30. Udagawa, K., Kaminaga, S., Tomioka, S., Yamasaki, H.: Two-dimensional numerical simulation on boundary layer control by MHD interaction. In: Proc. \(15^{\rm th}\) ICMHD 2, Moscow,591–596 (2005)

  31. Wang, Y.G., Zhu, S.Y.: Back flow of the two-dimensional unsteady Prandtl boundary layer under an adverse pressure gradient. SIAM J. Math. Anal. 52, 954–966 (2020)

    Article  MathSciNet  Google Scholar 

  32. Weinan, E.: Boundary layer theory and the zero-viscosity limit of the Navier–Stokes equation. Acta Math. Sin. (Engl. Ser.) 16, 207–218 (2000)

    Article  MathSciNet  Google Scholar 

  33. Weinan, E., Engquist, B.: Blow up of solutions of the unsteady Prandtl equation. Comm. Pure Appl. Math. 50, 1287–1293 (1997)

    Article  MathSciNet  Google Scholar 

  34. Xin, Z.P., Zhang, L.Q.: On the global existence of solutions to the Prandtl’s system. Adv. Math. 181, 88–133 (2004)

    Article  MathSciNet  Google Scholar 

  35. Xie, F., Yang, T.: Global-in-time stability of 2D MHD boundary layer in the Prandtl-Hartmann regime. SIAM J. Math. Anal. 50, 5749–5760 (2018)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to express their gratitude to Professor Ya-Guang Wang for the valuable discussion on this topic and instructive suggestions on the presentation, and to the referee for the valuable suggestions on improving this manuscript. This research was partially supported by National Natural Science Foundation of China (NNSFC) under Grant No. 11631008.

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Authors and Affiliations

  1. School of Mathematic Science, Shanghai Jiao Tong University, Shanghai, 200240, China

    Shengbo Gong & Xiang Wang

  2. HUAWEI Technologies Co., LTD. in Shanghai, Shanghai, 200210, China

    Shengbo Gong

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  1. Shengbo Gong
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  2. Xiang Wang
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Correspondence to Xiang Wang.

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Communicated by G.-Q. G. Chen

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Gong, S., Wang, X. On a Global Weak Solution and Back Flow of the Mixed Prandtl–Hartmann Boundary Layer Problem. J. Math. Fluid Mech. 23, 11 (2021). https://doi.org/10.1007/s00021-020-00530-6

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