A uniform estimate for the incompressible magneto-hydrodynamics equations with a slip boundary condition

Meng, Y.; Wang, Y.-G.

doi:10.1090/qam/1406

Authors: Y. Meng and Y.-G. Wang
Journal: Quart. Appl. Math. 74 (2016), 27-48
MSC (2010): Primary 35M13, 35Q35, 76D10, 76D03, 76N20
DOI: https://doi.org/10.1090/qam/1406
Published electronically: December 3, 2015
MathSciNet review: 3472518
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we derive a uniform estimate of the strong solution to the incompressible magneto-hydrodynamic (MHD) system with a slip boundary condition in a conormal Sobolev space with viscosity weight. As a consequence of this uniform estimate, we obtain that the solution of the viscous MHD system converges strongly to a solution of the ideal MHD system from a compactness argument.

References

R. Alexandre, Y.-G. Wang, C.-J. Xu, and T. Yang, Well-posedness of the Prandtl equation in Sobolev spaces, J. Amer. Math. Soc. 28 (2015), no. 3, 745–784. MR 3327535, DOI 10.1090/S0894-0347-2014-00813-4
Thierry Clopeau, Andro Mikelić, and Raoul Robert, On the vanishing viscosity limit for the $2\textrm {D}$ incompressible Navier-Stokes equations with the friction type boundary conditions, Nonlinearity 11 (1998), no. 6, 1625–1636. MR 1660366, DOI 10.1088/0951-7715/11/6/011
Weinan E and Bjorn Engquist, Blowup of solutions of the unsteady Prandtl’s equation, Comm. Pure Appl. Math. 50 (1997), no. 12, 1287–1293. MR 1476316, DOI 10.1002/(SICI)1097-0312(199712)50:12<1287::AID-CPA4>3.0.CO;2-4
David Gérard-Varet and Emmanuel Dormy, On the ill-posedness of the Prandtl equation, J. Amer. Math. Soc. 23 (2010), no. 2, 591–609. MR 2601044, DOI 10.1090/S0894-0347-09-00652-3
Emmanuel Grenier, On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math. 53 (2000), no. 9, 1067–1091. MR 1761409, DOI 10.1002/1097-0312(200009)53:9<1067::AID-CPA1>3.3.CO;2-H
Olivier Guès, Problème mixte hyperbolique quasi-linéaire caractéristique, Comm. Partial Differential Equations 15 (1990), no. 5, 595–645 (French). MR 1070840, DOI 10.1080/03605309908820701
Olivier Guès, Guy Métivier, Mark Williams, and Kevin Zumbrun, Existence and stability of noncharacteristic boundary layers for the compressible Navier-Stokes and viscous MHD equations, Arch. Ration. Mech. Anal. 197 (2010), no. 1, 1–87. MR 2646814, DOI 10.1007/s00205-009-0277-y
Dragoş Iftimie and Gabriela Planas, Inviscid limits for the Navier-Stokes equations with Navier friction boundary conditions, Nonlinearity 19 (2006), no. 4, 899–918. MR 2214949, DOI 10.1088/0951-7715/19/4/007
Dragoş Iftimie and Franck Sueur, Viscous boundary layers for the Navier-Stokes equations with the Navier slip conditions, Arch. Ration. Mech. Anal. 199 (2011), no. 1, 145–175. MR 2754340, DOI 10.1007/s00205-010-0320-z
Tosio Kato, Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary, Seminar on nonlinear partial differential equations (Berkeley, Calif., 1983) Math. Sci. Res. Inst. Publ., vol. 2, Springer, New York, 1984, pp. 85–98. MR 765230, DOI 10.1007/978-1-4612-1110-5_{6}
James P. Kelliher, Navier-Stokes equations with Navier boundary conditions for a bounded domain in the plane, SIAM J. Math. Anal. 38 (2006), no. 1, 210–232. MR 2217315, DOI 10.1137/040612336
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969 (French). MR 0259693
M. C. Lopes Filho, A. L. Mazzucato, H. J. Nussenzveig Lopes, and Michael Taylor, Vanishing viscosity limits and boundary layers for circularly symmetric 2D flows, Bull. Braz. Math. Soc. (N.S.) 39 (2008), no. 4, 471–513. MR 2465261, DOI 10.1007/s00574-008-0001-9
M. C. Lopes Filho, H. J. Nussenzveig Lopes, and G. Planas, On the inviscid limit for two-dimensional incompressible flow with Navier friction condition, SIAM J. Math. Anal. 36 (2005), no. 4, 1130–1141. MR 2139203, DOI 10.1137/S0036141003432341
Nader Masmoudi and Frédéric Rousset, Uniform regularity for the Navier-Stokes equation with Navier boundary condition, Arch. Ration. Mech. Anal. 203 (2012), no. 2, 529–575. MR 2885569, DOI 10.1007/s00205-011-0456-5
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), no. 10, 1683–1741. DOI 10.1002/cpa.21595.
Yasunori Maekawa, On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half-plane, Comm. Pure Appl. Math. 67 (2014), no. 7, 1045–1128. MR 3207194, DOI 10.1002/cpa.21516
C. L. M. H. Navier, Sur les lois de l’équilibrie et du mouvement des corps élastiques, Mem. Acad. R. Sci. Inst. France 6 (1827), 369.
O. A. Oleĭnik, On properties of solutions of certain boundary problems for equations of elliptic type, Mat. Sbornik N.S. 30(72) (1952), 695–702 (Russian). MR 0050125
O. A. Oleinik and V. N. Samokhin, Mathematical models in boundary layer theory, Applied Mathematics and Mathematical Computation, vol. 15, Chapman & Hall/CRC, Boca Raton, FL, 1999. MR 1697762
L. Prandtl, Über flüssigkeitsbewegungen bei sehr kleiner Reibung, in Verh. Int. Math. Kongr., Heidelberg, Germany, 1904, Teubner, Germany, 1905, pp. 484-494.
Marco Sammartino and Russel E. Caflisch, 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. 192 (1998), no. 2, 463–491. MR 1617538, DOI 10.1007/s002200050305
Michel Sermange and Roger Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math. 36 (1983), no. 5, 635–664. MR 716200, DOI 10.1002/cpa.3160360506
Roger Temam and Xiaoming Wang, On the behavior of the solutions of the Navier-Stokes equations at vanishing viscosity, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 3-4, 807–828 (1998). Dedicated to Ennio De Giorgi. MR 1655543
Xiao-Ping Wang, Ya-Guang Wang, and Zhouping Xin, Boundary layers in incompressible Navier-Stokes equations with Navier boundary conditions for the vanishing viscosity limit, Commun. Math. Sci. 8 (2010), no. 4, 965–998. MR 2744916
Yuelong Xiao and Zhouping Xin, On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition, Comm. Pure Appl. Math. 60 (2007), no. 7, 1027–1055. MR 2319054, DOI 10.1002/cpa.20187
Yuelong Xiao, Zhouping Xin, and Jiahong Wu, Vanishing viscosity limit for the 3D magnetohydrodynamic system with a slip boundary condition, J. Funct. Anal. 257 (2009), no. 11, 3375–3394. MR 2571431, DOI 10.1016/j.jfa.2009.09.010
Zhouping Xin and Liqun Zhang, On the global existence of solutions to the Prandtl’s system, Adv. Math. 181 (2004), no. 1, 88–133. MR 2020656, DOI 10.1016/S0001-8708(03)00046-X
V. I. Judovič, A two-dimensional non-stationary problem on the flow of an ideal incompressible fluid through a given region, Mat. Sb. (N.S.) 64 (106) (1964), 562–588 (Russian). MR 0177577