Approximations to Isentropic Planar Magneto-Hydrodynamics Equations by Relaxed Euler-Type Systems

Abstract

In this paper, the authors consider an approximation to the isentropic planar Magneto-hydrodynamics (MHD for short) equations by a kind of relaxed Euler-type system. The approximation is based on the generalization of the Maxwell law for non-Newtonian fluids together with the Maxwell correction for the Ampère law, hence the approximate system becomes a first-order quasilinear symmetrizable hyperbolic systems with partial dissipation. They establish the global-in-time smooth solutions to the approximate Euler-type equations in a small neighbourhood of constant equilibrium states and obtain the global-in-time convergence towards the isentropic planar MHD equations. In addition, they also establish the global-in-time error estimates of the limit based on stream function techniques and energy estimates for error variables.

Keywords

Planar MHD equations / Relaxation limits / Global convergence / Stream function

References

[1] Bresch D, Prange C. Newtonian limit for weakly viscoelastic fluid flows. SIAM J. Math. Anal., 2014, 46(2): 1116-1159

[2] Cattaneo O. Sulla conduzione del calore. Atti Sem. Mat. Fis. Univ. Modena, 1949, 3: 83-101

[3] Cattaneo C. Sur une forme de l’équation de la chaleur éliminant le paradoxe d’une propagation instantanée. C. R. Acad. Sci. Paris, 1958, 247: 431-433

[4] Chen F. Introduction to Plasma Physics and Controlled Fusion, 1, 1984, New York: PlenumPress

[5] Chen G-Q, Wang D. Global solutions of nonlinear magnetohydrodynamics with large initial data. J. Differential Equations, 2002, 182(2): 344-376

[6] Fernández Sare H D, Racke R. On the stability of damped Timoshenko systems: Cattaneo versus Fourier law. Arch. Ration. Mech. Anal., 2009, 194(1): 221-251

[7] Hu Y, Ju Q. Global large solutions of magnetohydrodynamics with temperature-dependent heat conductivity. Z. Angew. Math. Phys., 2015, 66(3): 865-889

[8] Hu Y, Racke R. Compressible Navier-Stokes equations with hyperbolic heat conduction. J. Hyperbolic Differ. Equ., 2016, 13(2): 233-247

[9] Hu Y, Racke R. Compressible Navier-Stokes equations with revised Maxwell’s law. J. Math. Fluid Mech., 2017, 19(1): 77-90

[10] Hu Y, Racke R. Hyperbolic compressible Navier-Stokes equations. J. Differential Equations, 2020, 269(4): 3196-3220

[11] Hu Y, Racke R, Wang N. Formation of singularities for one-dimensional relaxed compressible Navier-Stokes equations. J. Differential Equations, 2022, 327: 145-165

[12] Kato T. The Cauchy problem for quasi—linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal., 1975, 58(3): 181-205

[13] Kawashima, S., Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, PhD thesis, Kyoto Uninversity, 1983.

[14] Kawashima S, Okada M. Smooth global solutions for the one-dimensional equations in magnetohydrodynamics. Proc. Japan Acad. Ser. A Math. Sci., 1982, 58(9): 384-387

[15] Lax P D. Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, 11, 1973, Philadelphia, Pa.: Society for Industrial and Applied Mathematics

[16] Li T, Qin T. Physics and Partial Differential Equations, 1, 2012, Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) Higher Education Press, Beijing

[17] Li, W., Shang, Z. and Tang, F., Global existence of large solutions to the planar magnetohydrodynamic equations with zero magnetic diffusivity, J. Math. Anal. Appl., 496 (1), 2021, 31 pp.

[18] Li Y. Global well-posedness to the one-dimensional model for planar non-resistive magnetohydrodynamics with large data and vacuum. J. Math. Anal. Appl., 2018, 462(2): 1342-1356

[19] Li Y, Peng Y-J, Zhao L. Convergence rate from hyperbolic systems of balance laws to parabolic systems. Appl. Anal., 2021, 100(5): 1079-1095

[20] Li Y, Peng Y-J, Zhao L. Convergence rates in zero-relaxation limits for Euler-Maxwell and Euler-Poisson systems. J. Math. Pures Appl. (9), 2021, 154: 185-211

[21] Li Y, Shang Z. Global large solutions to planar magnetohydrodynamics equations with temperature-dependent coefficients. J. Hyperbolic Differ. Equ., 2019, 16(3): 443-493

[22] Liu X, Qin Y. Global solutions to isentropic planar magnetohydrodynamic equations with density-dependent viscosity. Math. Methods Appl. Sci., 2018, 41(12): 4448-4464

[23] Majda A. Compressible fluid flow and systems of conservation laws in several space variables, 53, 1984, New York: Springer-Verlag

[24] Maxwell J C IV. On the dynamical theory of gases. Philosophical Transactions of the Royal Society of London, 1867, 157: 49-88

[25] Molinet L, Talhouk R. Newtonian limit for weakly viscoelastic fluid flows of Oldroyd type. SIAM J. Math. Anal., 2008, 39(5): 1577-1594

[26] Ou Y, Shi P, Wittwer P. Large time behaviors of strong solutions to magnetohydrodynamic equations with free boundary and degenerate viscosity. J. Math. Phys., 2018, 59(8): 081510 34 pp

[27] Peng Y-J. Relaxed Euler systems and convergence to Navier-Stokes equations. Ann. Inst. H. Poincaré C Anal. Non Linéaire, 2021, 38(2): 369-401

[28] Peng Y-J, Wasiolek V. Parabolic limit with differential constraints of first—order quasilinear hyperbolic systems. Ann. Inst. H. Poincaré Anal. Non Linéaire, 2016, 33(4): 1103-1130

[29] Peng Y-J, Wasiolek V. Uniform global existence and parabolic limit for partially dissipative hyperbolic systems. J. Differential Equations, 2016, 260(9): 7059-7092

[30] Peng, Y.-J. and Zhao, L., Global convergence to compressible full Navier-Stokes equations by approximation with Oldroyd-type constitutive laws, J. Math. Fluid Mech., 24 (2), 2022, 17 pp.

[31] Said-Houari B, Kasimov A. Damping by heat conduction in the Timoshenko system: Fourier and Cattaneo are the same. J. Differential Equations, 2013, 255(4): 611-632

[32] Saut J-C. Some remarks on the limit of viscoelastic fluids as the relaxation time tends to zero, Trends in applications of pure mathematics to mechanics (Bad Honnef, 1985), 249, 1986, Berlin: Springer-Verlog 364-369

[33] Shang Z. Global large solutions to the Cauchy problem of planar magnetohydrodynamics equations with temperature-dependent coefficients. J. Dyn. Control Syst., 2022, 28(1): 163-205

[34] Shizuta Y, Kawashima S. Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation. Hokkaido Math. J., 1985, 14(2): 249-275

[35] Simon J. Compact sets in the space L ^p(0, T; B). Ann. Mat. Pura Appl. (4), 1987, 146: 65-96

[36] Smagulov S S, Durmagambetov A A, Iskenderova D A. Cauchy problems for equations of magnetogasdynamics. Differentsial’ nye Uravneniya, 1993, 29(2): 337-348 367 pp

[37] Tao, Q., Yang, Y. and Gao, J., A free boundary problem for planar compressible Hall-magnetohydrodynamic equations, Z. Angew. Math. Phys., 69 (1), 2018, 22 pp.

[38] Tao Q, Yang Y, Yao Z-A. Global existence and exponential stability of solutions for planar compressible Hall-magnetohydrodynamic equations. J. Differential Equations, 2017, 263(7): 3788-3831

[39] Wang D. Large solutions to the initial-boundary value problem for planar magnetohydrodynamics. SIAM J. Appl. Math., 2003, 63(4): 1424-1441

[40] Ye X, Zhang J. On the behavior of boundary layers of one-dimensional isentropic planar MHD equations with vanishing shear viscosity limit. J. Differential Equations, 2016, 260(4): 3927-3961

[41] Yong W-A. Newtonian limit of Maxwell fluid flows. Arch. Ration. Mech. Anal., 2014, 214(3): 913-922

[42] Zhao L, Xi S. Convergence rate from systems of balance laws to isotropic parabolic systems, a periodic case. Asymptot. Anal., 2021, 124: 163-198