A uniform estimate for the incompressible magneto-hydrodynamics equations with a slip boundary condition
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
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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
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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
- T. Clopeau, A. Mikelić, and R. 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 (99g:35102), DOI 10.1088/0951-7715/11/6/011
- W. E and B. Engquist, Blowup of solutions of the unsteady Prandtl’s equation, Comm. Pure Appl. Math. 50 (1997), no. 12, 1287–1293. MR 1476316 (99c:35196), DOI 10.1002/(SICI)1097-0312(199712)50:12$\langle$1287::AID-CPA4$\rangle$3.0.CO;2-4
- D. Gérard-Varet and E. Dormy, On the ill-posedness of the Prandtl equation, J. Amer. Math. Soc. 23 (2010), no. 2, 591–609. MR 2601044 (2011f:35259), DOI 10.1090/S0894-0347-09-00652-3
- E. Grenier, On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math. 53 (2000), no. 9, 1067–1091. MR 1761409 (2001i:76056), DOI 10.1002/1097-0312(200009)53:9$\langle$1067::AID-CPA1$\rangle$3.3.CO;2-H
- O. Guès, Problème mixte hyperbolique quasi-linéaire caractéristique, Comm. Partial Differential Equations 15 (1990), no. 5, 595–645 (French). MR 1070840 (91i:35122), DOI 10.1080/03605309908820701
- O. Guès, G. Métivier, M. Williams, and K. 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 (2012f:35400), DOI 10.1007/s00205-009-0277-y
- D. Iftimie and G. Planas, Inviscid limits for the Navier-Stokes equations with Navier friction boundary conditions, Nonlinearity 19 (2006), no. 4, 899–918. MR 2214949 (2007c:35130), DOI 10.1088/0951-7715/19/4/007
- D. Iftimie and F. 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 (2012b:76031), DOI 10.1007/s00205-010-0320-z
- T. 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 (86a:35116), DOI 10.1007/978-1-4612-1110-5_6
- J. 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 (electronic). MR 2217315 (2007a:35118), 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 (41 \#4326)
- M. C. Lopes Filho, A. L. Mazzucato, H. J. Nussenzveig Lopes, and M. 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 (2010b:35344), 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 (electronic). MR 2139203 (2005k:76026), DOI 10.1137/S0036141003432341
- N. Masmoudi and F. 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.
- Y. 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 (14,280a)
- 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 (2000c:76021)
- L. Prandtl, Über flüssigkeitsbewegungen bei sehr kleiner Reibung, in Verh. Int. Math. Kongr., Heidelberg, Germany, 1904, Teubner, Germany, 1905, pp. 484-494.
- M. Sammartino and R. E. Caflisch, 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. 192 (1998), 433–461. MR1617542 (99d:35129a); II. Construction of the Navier-Stokes solution, Comm. Math. Phys. 192 (1998), no. 2, 463–491. MR 1617538 (99d:35129b), DOI 10.1007/s002200050305
- M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math. 36 (1983), no. 5, 635–664. MR 716200 (85k:76042), DOI 10.1002/cpa.3160360506
- R. Temam and X. Wang, On the behavior of the solutions of the Navier-Stokes equations at vanishing viscosity, dedicated to Ennio De Giorgi, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 3-4, 807–828 (1998). MR 1655543 (99j:35169)
- X.-P. Wang, Y.-G. Wang, and Z. 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 (2011k:35173)
- Y. Xiao and Z. 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 (2009g:35224), DOI 10.1002/cpa.20187
- Y. Xiao, Z. Xin, and J. 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 (2010k:35388), DOI 10.1016/j.jfa.2009.09.010
- Z. Xin and L. Zhang, On the global existence of solutions to the Prandtl’s system, Adv. Math. 181 (2004), no. 1, 88–133. MR 2020656 (2005f:35219), DOI 10.1016/S0001-8708(03)00046-X
- V. I. Judovič [Yudovich], 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 (31 \#1840)
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Additional Information
Y. Meng
Affiliation:
Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China — and — School of Mathematics and Physics, Jiangsu University of Science and Technology, Zhenjiang, Jiangsu, 212003, People’s Republic of China
Email:
myp_just@163.com
Y.-G. Wang
Affiliation:
Department of Mathematics, MOE-LSC and SHL-MAC, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China
MR Author ID:
291072
Email:
ygwang@sjtu.edu.cn
Keywords:
Incompressible MHD equations,
uniform estimate,
conormal Sobolev spaces,
small viscosity limit.
Received by editor(s):
March 29, 2014
Published electronically:
December 3, 2015
Additional Notes:
The first author was supported by the Shanghai Jiao Tong University Innovation Fund for Postgraduates and Scientific Research Fund of Jiangsu University of Science and Technology
This work was partially supported by NNSF of China under the grants 10971134, 11031001, 91230102, and by Shanghai Committee of Science and Technology under the grant 15XD1502300
Article copyright:
© Copyright 2015
Brown University