Abstract
In this paper, we study the low Mach number limit of the compressible Hall-magnetohydrodynamic equations. It is justified rigorously that, for the well-prepared initial data, the classical solutions of the compressible Hall-magnetohydrodynamic equations converge to that of the incompressible Hall-magnetohydrodynamic equations as the Mach number tends to zero.
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References
Acheritogaray M., Degond P., Frouvelle A., Liu J.-G.: Kinetic fomulation and glabal existence for the Hall-magnetohydrodynamics system. Kinet. Rel. Models 4, 901–918 (2011)
Chae D., Lee J.: On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics. J. Differ. Equ. 256, 3835–3858 (2014)
Chae D., Degond P., Liu J.-G.: Well-posedness for Hall-magnetohydrodynamics. Ann. Inst. H. Poincaré Anal. Non Linéaire 31, 555–565 (2014)
Chae D., Schonbek M.: On the temporal decay for the Hall-magnetohydrodynamic equations. J. Differ. Equ. 255(11), 3971–3982 (2013)
Chae, D., Weng, S.: Singularity formation for the incompressible Hall-MHD equations without resistivity. Ann. Inst. H. Poincaré Anal. Non Linéaire (published online)
Dou C.-S., Jiang S., Ju Q.-C.: Global existence and the low Mach number limit for the compressible magnetohydrodynamic equations in a bounded domain with perfectly conducting boundary. Z. Angew. Math. Phys. 64(6), 1661–1678 (2013)
Dreher J., Runban V., Grauer R.: Axisymmetric flows in Hall-MHD: a tendency towards finite-time singularity formation. Phys. Scr. 72, 451–455 (2005)
Dumas E., Sueur F.: On the weak solutions to the Maxwell–Landau–Lifshitz equations and to the Hall-magnetohydrodynamics equations. Commun. Math. Phys. 330(3), 1179–1225 (2014)
Fan J.-S., Huang S.-X., Nakamura G.: Well-posedness for the axisymmetric incompressible viscous Hall-magnetohydrodynamics equations. Appl. Math. Lett. 26(9), 963–967 (2013)
Fan J.-S., Li F.-C., Nakamura G.: Regularity criteria for the incompressible Hall-magnetohydrodynamic equations. Nonlinear Anal. 109, 173–179 (2014)
Fan J.-S., Alsaedi A., Hayat T., Nakamura G., Zhou Y.: On strong solutions to the compressible Hall-magnetohydrodynamic system. Nonlinear Anal. Real World Appl. 22, 423–434 (2015)
Fan J.-S., Ozawa T.: Regularity criteria for Hall-magnetohydrodynamics and the space-time monopole equation in Lorenz gauge. Contemp. Math. 612, 81–89 (2014)
Feireisl E., Novotny A., Sun Y.: Dissipative solutions and the incompressible inviscid limits of the compressible magnetohydrodynamic system in unbounded domains. Discret. Contin. Dyn. Syst. 34(1), 121–143 (2014)
Hu X.-P., Wang D.-H.: Global solutions to the three-dimensional full compressible magnetohydrodynamic flows. Commun. Math. Phys. 283, 255–284 (2008)
Hu X.-P., Wang D.-H.: Low Mach number limit of viscous compressible magnetohydrodynamic flows. SIAM J. Math. Anal. 41, 1272–1294 (2009)
Jiang S., Ju Q.-C., Li F.-C.: Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions. Commun. Math. Phys. 297, 371–400 (2010)
Jiang S., Ju Q.-C., Li F.-C.: Incompressible limit of the compressible magnetohydrodynamic equations with vanishing viscosity coefficients. SIAM J. Math. Anal. 42, 2539–2553 (2010)
Jiang S., Ju Q.-C., Li F.-C.: Low Mach number limit for the multi-dimensional full magnetohydrodynamic equations. Nonlinearity 25, 1351–1365 (2012)
Klainerman S., Majda A.: Singular perturbations of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math. 34, 481–524 (1981)
Li F.-C., Mu Y.-M.: Low Machnumber limit of the full compressible Navier–Stokes–Maxwell system. J. Math. Anal. Appl. 412, 334–344 (2014)
Li F.-C., Yu H.-Y.: Optimal decay rate of classical solutions to the compressible magnetohydrodynamic equations. Proc. R. Soc. Edinb. Sect. A 141(1), 109–126 (2011)
Li X.-L., Su N., Wang D.-H.: Local strong solution to the compressible magnetohydrodynamic flow with large data. J. Hyperbolic Differ. Equ. 8(3), 415–436 (2011)
Li Y.-P.: Convergence of the compressible magnetohydrodynamic equations to incompressible magnetohydrodynamic equations. J. Differ. Equ. 252, 2725–2738 (2012)
Polygiannakis J.M., Moussas X.: A review of magneto-vorticity induction in Hall-MHD plasmas. Plasma Phys. Control. Fusion 43, 195–221 (2001)
Suen A., Hoff D.: Global low-energy weak solutions of the equations of three-dimensional compressible magnetohydrodynamics. Arch. Ration. Mech. Anal. 205(1), 27–58 (2012)
Volpert, A.I., Hudjaev S.I.: On the Cauchy problem for composite systems of nonlinear differential equations. Mat. Sb. 87, 504–528 (1972) (Russian) [English transl.: Math. USSR Sb. 16, 517–544 (1973)]
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Mu, Y. Zero Mach number limit of the compressible Hall-magnetohydrodynamic equations. Z. Angew. Math. Phys. 67, 1 (2016). https://doi.org/10.1007/s00033-015-0604-0
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DOI: https://doi.org/10.1007/s00033-015-0604-0