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Analytical study of certain magnetohydrodynamic-\(\alpha\) models. (English) Zbl 1144.81378
Summary: In this paper we present an analytical study of a subgrid scale turbulence model of the three-dimensional magnetohydrodynamic (MHD) equations, inspired by the Navier-Stokes-alpha (also known as the viscous Camassa-Holm equations or the Lagrangian-averaged Navier-Stokes-alpha model). Specifically, we show the global well-posedness and regularity of solutions of a certain MHD-alpha model (which is a particular case of the Lagrangian averaged magnetohydrodynamic-alpha model without enhancing the dissipation for the magnetic field). We also introduce other subgrid scale turbulence models, inspired by the Leray-alpha and the modified Leray-alpha models of turbulence. Finally, we discuss the relation of the MHD-alpha model to the MHD equations by proving a convergence theorem, that is, as the length scale alpha tends to zero, a subsequence of solutions of the MHD-alpha equations converges to a certain solution (a Leray-Hopf solution) of the three-dimensional MHD equations.

MSC:
76W05 Magnetohydrodynamics and electrohydrodynamics
35Q35 PDEs in connection with fluid mechanics
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76F02 Fundamentals of turbulence
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References:
[1] DOI: 10.1103/PhysRevLett.81.5338 · Zbl 1042.76525 · doi:10.1103/PhysRevLett.81.5338
[2] DOI: 10.1016/S0167-2789(99)00098-6 · Zbl 1194.76069 · doi:10.1016/S0167-2789(99)00098-6
[3] DOI: 10.1063/1.870096 · Zbl 1147.76357 · doi:10.1063/1.870096
[4] DOI: 10.1098/rspa.2004.1373 · Zbl 1145.76386 · doi:10.1098/rspa.2004.1373
[5] DOI: 10.1007/s002200000349 · Zbl 0988.76020 · doi:10.1007/s002200000349
[6] DOI: 10.1080/01630569308816523 · Zbl 0792.35096 · doi:10.1080/01630569308816523
[7] DOI: 10.1063/1.868526 · Zbl 1023.76513 · doi:10.1063/1.868526
[8] DOI: 10.1007/BF00250512 · Zbl 0264.73027 · doi:10.1007/BF00250512
[9] DOI: 10.1080/03605309808821336 · doi:10.1080/03605309808821336
[10] DOI: 10.1016/S0167-2789(01)00191-9 · Zbl 1037.76022 · doi:10.1016/S0167-2789(01)00191-9
[11] DOI: 10.1023/A:1012984210582 · Zbl 0995.35051 · doi:10.1023/A:1012984210582
[12] DOI: 10.1016/0022-1236(89)90015-3 · Zbl 0702.35203 · doi:10.1016/0022-1236(89)90015-3
[13] DOI: 10.1063/1.1529180 · Zbl 1185.76144 · doi:10.1063/1.1529180
[14] DOI: 10.1016/S0167-2789(99)00093-7 · Zbl 1194.76062 · doi:10.1016/S0167-2789(99)00093-7
[15] DOI: 10.1016/S0167-2789(02)00552-3 · Zbl 1098.76547 · doi:10.1016/S0167-2789(02)00552-3
[16] DOI: 10.1063/1.1460941 · Zbl 1080.76504 · doi:10.1063/1.1460941
[17] DOI: 10.1088/0305-4470/35/3/313 · Zbl 1040.76054 · doi:10.1088/0305-4470/35/3/313
[18] DOI: 10.1006/aima.1998.1721 · Zbl 0951.37020 · doi:10.1006/aima.1998.1721
[19] DOI: 10.1175/1520-0485(2003)033<2355:MMTITB>2.0.CO;2 · doi:10.1175/1520-0485(2003)033<2355:MMTITB>2.0.CO;2
[20] DOI: 10.1088/0951-7715/19/4/006 · Zbl 1106.35050 · doi:10.1088/0951-7715/19/4/006
[21] DOI: 10.1201/9780203017692.ch10 · doi:10.1201/9780203017692.ch10
[22] DOI: 10.1007/BF00312444 · Zbl 0838.76066 · doi:10.1007/BF00312444
[23] Ladyzhenskaya O. A., Proc. Steklov Inst. Math. 102 pp 95– (1967)
[24] DOI: 10.1007/978-1-4757-4317-3 · doi:10.1007/978-1-4757-4317-3
[25] DOI: 10.1016/S0893-9659(03)90118-2 · Zbl 1039.76027 · doi:10.1016/S0893-9659(03)90118-2
[26] Layton W., Discrete Contin. Dyn. Syst., Ser. B 6 pp 111– (2006)
[27] DOI: 10.1007/s00021-005-0181-7 · Zbl 1099.76027 · doi:10.1007/s00021-005-0181-7
[28] Lions J. L., Bull. Soc. Math. France 87 pp 245– (1959)
[29] Lions J. L., Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires (1969)
[30] DOI: 10.1007/s00205-002-0207-8 · Zbl 1020.76014 · doi:10.1007/s00205-002-0207-8
[31] Metivier G., J. Math. Pures Appl. 57 pp 133– (1978)
[32] DOI: 10.1103/PhysRevE.71.046304 · doi:10.1103/PhysRevE.71.046304
[33] DOI: 10.1063/1.1863260 · Zbl 1187.76356 · doi:10.1063/1.1863260
[34] DOI: 10.1063/1.1533069 · Zbl 1185.76263 · doi:10.1063/1.1533069
[35] DOI: 10.1090/trans2/026/05 · Zbl 0131.31803 · doi:10.1090/trans2/026/05
[36] DOI: 10.1063/1.2194966 · Zbl 1185.76714 · doi:10.1063/1.2194966
[37] DOI: 10.1002/cpa.3160360506 · Zbl 0524.76099 · doi:10.1002/cpa.3160360506
[38] DOI: 10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2 · doi:10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2
[39] DOI: 10.1007/978-1-4612-0873-0 · doi:10.1007/978-1-4612-0873-0
[40] Temam R., Studies in Mathematics and its Applications 2, in: Navier-Stokes Equations. Theory and Numerical Analysis, 3. ed. (1984) · Zbl 0568.35002
[41] DOI: 10.1137/1.9781611970050 · doi:10.1137/1.9781611970050
[42] Vishik M. I., Sov. Math. Dokl. 71 pp 92– (2005)
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