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On the incompressible limits for the full magnetohydrodynamics flows. (English) Zbl 1242.35051
This paper studies the incompressible limits for weak solutions for the full magentohydrodynamics flows in bounded and unbounded domains. In the model, various physically acceptable assumptions are made, e.g., the viscous stress tension is determined through Newton’s rheological law, the heat flux is given by Fourier’s law etc., and a scaling of the dimensionless parameters of the Mach, Froude and Alfven number is assumed according to which $Ma=ϵ$, $Fr=\sqrt{ϵ}$, $Al=\sqrt{ϵ}$ where $ϵ$ is small. A variational formulation is provided for the full problem and specific conditions for the data of the problem so that this formulation holds are stated. The limit as $ϵ\to 0$ is studied in great detail in both bounded and unbounded domains.
##### MSC:
 35B40 Asymptotic behavior of solutions of PDE 76N10 Compressible fluids, general 35B45 A priori estimates for solutions of PDE 76W05 Magnetohydrodynamics and electrohydrodynamics 35Q35 PDEs in connection with fluid mechanics 35B25 Singular perturbations (PDE)
##### References:
 [1] Ducomet, B.; Feireisl, E.: The equations of magnetohydrodynamics: on the interaction between matter and radiation in the evolution of gaseous stars, Comm. math. Phys. 266, 595-629 (2006) · Zbl 1113.76098 · doi:10.1007/s00220-006-0052-y [2] Eliezer, S.; Ghatak, A.; Hora, H.: An introduction to equations of states, theory and applications, (1986) [3] Feireisl, E.: Incompressible limits and propagation of acoustic waves in large domains with boundaries, Comm. math. Phys. 294, No. 1, 73-95 (2010) · Zbl 1208.35110 · doi:10.1007/s00220-009-0954-6 [4] Feireisl, E.: Stability of flows of real monoatomic gases, Comm. partial differential equations 31, 325-348 (2006) · Zbl 1092.35077 · doi:10.1080/03605300500358186 [5] Feireisl, E.; Novotný, A.: Singular limits in thermodynamics of viscous fluids, Adv. math. Fluid mech. (2009) [6] Feireisl, E.; Novotný, A.: The low Mach number limit for the full Navier-Stokes-Fourier system, Arch. ration. Mech. anal. 186, No. 1, 77-107 (2007) · Zbl 1147.76049 · doi:10.1007/s00205-007-0066-4 [7] Feireisl, E.; Novotný, A.; Petzeltová, H.: On the incompressible limit for the Navier-Stokes-Fourier system in domains with wavy bottoms, Math. models methods appl. Sci. 18, 291-324 (2008) · Zbl 1158.35072 · doi:10.1142/S0218202508002681 [8] Hu, Xianpeng; Wang, Dehua: Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. math. Phys. 283, 255-284 (2008) · Zbl 1158.35075 · doi:10.1007/s00220-008-0497-2 [9] Kato, T.: Wave operators and similarity for some non-selfadjoint operators, Math. ann. 162, 258-279 (1965/1966) · Zbl 0139.31203 · doi:10.1007/BF01360915 [10] Klainerman, S.; Majda, A.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. pure appl. Math. 34, 481-524 (1981) · Zbl 0476.76068 · doi:10.1002/cpa.3160340405 [11] Peter Kukucka, Singular limits of the equations of magnetohydrodynamics, Preprint. [12] Klein, R.; Botta, N.; Schneider, T.; Munz, C. D.; Roller, S.; Meister, A.; Hoffmann, L.; Sonar, T.: Asymptotic adaptive methods for multi-scale problems in fluid mechanics, J. engrg. Math. 39, 261-343 (2001) · Zbl 1015.76071 · doi:10.1023/A:1004844002437 [13] Lions, P. -L.; Masmoudi, N.: Incompressible limit for a viscous compressible fluid, J. math. Pures appl. (9) 77, 585-627 (1998) · Zbl 0909.35101 · doi:10.1016/S0021-7824(98)80139-6 [14] Oxenius, J.: Kinetic theory of particles and photons, (1986) [15] Poul, L.: Existence of weak solutions to the Navier-Stokes-Fourier system on Lipschitz domains, Discrete contin. Dyn. syst. (Suppl.), 834-843 (2007) · Zbl 1163.35456 · doi:http://www.aimsciences.org/journals/redirecting.jsp?paperID=2894 [16] Reed, M.; Simon, B.: Methods of modern mathematical physics. IV. analysis of operator, (1978) [17] Vishik, M. I.; Ljusternik, L. A.: Regular perturbations and a boundary layer for linear differential equations with a small parameter, Uspekhi mat. Nauk 12, 3-122 (1957) · Zbl 0087.29602