Singular limits of the equations of compressible ideal magneto-hydrodynamics with physical boundary conditions are considered. The uniform existence of classical solutions with respect to the Mach number and Alfvén number is established by energy methods. Under appropriate conditions on the initial data, convergence of the solutions of the original system is proved as the two small parameters tend to zero with the limiting solutions satisfying the two-dimensional incompressible flow.
T.Alazard, Incompressible limit of the nonisentropic Euler equations with the solid wall boundary conditions, Adv. Differential Equations10(1) (2005), 19–44.
2.
G.Browning and H.-O.Kreiss, Problems with different time scales for nonlinear partial differential equations, SIAM J. Appl. Math.42 (1982), 704–718. doi:10.1137/0142049.
3.
B.Cheng, Q.Ju and S.Schochet, Three-scale singular limits of evolutionary PDEs, Archive for Rational Mechanics and Analysis229(2) (2018), 601–625. doi:10.1007/s00205-018-1233-5.
4.
S.Goto, Singular limits of the incompressible ideal magneto-fluid motion with respect to the Alfvén number, Hokk. Math. J.19 (1990), 175–187. doi:10.14492/hokmj/1381517168.
5.
H.Isozaki, Singular limits for the compressible Euler equation in an exterior domain, J. Reine Angew. Math.381 (1987), 1–36.
6.
S.Jiang, Q.C.Ju and F.C.Li, Incompressible limit of the nonisentropic ideal magnetohydrodynamic equations, SIAM J. Math. Anal.48(1) (2016), 302–319. doi:10.1137/15M102842X.
7.
S.Klainerman and A.Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Commun. Pure Appl. Math.34 (1981), 481–524. doi:10.1002/cpa.3160340405.
8.
S.Klainerman and A.Majda, Compressible and incompressible fluids, Commun. Pure Appl. Math.35 (1982), 629–651. doi:10.1002/cpa.3160350503.
9.
T.Li and T.Qin, Physics and Partial Differential Equations, The Complete Set, SIAM, 2014.
10.
B.Rubino, Singular limits in the data space for the equations of magnetofluid dynamics, Hokkaido Math. J.24 (1995), 357–386. doi:10.14492/hokmj/1380892599.
11.
S.Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit, Comm. Math. Phys.104 (1986), 49–75. doi:10.1007/BF01210792.
12.
S.Schochet, Singular limits in bounded domains for quasilinear symmetric hyperbolic systems having a vorticity equation, J. Differential Equations68 (1987), 400–428. doi:10.1016/0022-0396(87)90178-1.
13.
J.Simon, Compact sets in the space , Ann. Math. Pura Appl.146 (1987), 65–96. doi:10.1007/BF01762360.
14.
T.Yanagisawa, The initial boundary value problem for the equations of ideal magnetohydrodynamics, Hokk. Math. J.16 (1987), 295–314. doi:10.14492/hokmj/1381518181.
15.
T.Yanagisawa and A.Matsumura, The fixed boundary value problems for the equations of ideal magneto-hydrodynamics with a perfectly conducting wall condition, Comm. Math. Phys.136 (1991), 119–140. doi:10.1007/BF02096793.
