Dissipative measure-valued solutions to the magnetohydrodynamic equations
Abstract
In this paper, we study the dissipative measure-valued solution to the magnetohydrodynamic equations of 3D compressible isentropic flows with the adiabatic exponent γ > 1 and prove that a dissipative measure-valued solution is the same as the standard smooth classical solution as long as the latter exists, provided they emanate from the same initial data (weak–strong) uniqueness principle.
Keyword : compressible magnetohydrodynamic equations, measure-valued solution, weak-strong uniqueness
How to Cite
Yang, J., Wang, H., & Shi, Q. (2025). Dissipative measure-valued solutions to the magnetohydrodynamic equations. Mathematical Modelling and Analysis, 30(1), 17–35. https://doi.org/10.3846/mma.2025.19998
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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B.-K. Huang. On the existence of dissipative measure-valued solutions to the compressible micropolar system. Journal of Mathematical Fluid Mechanics, 22(59), 2020. https://doi.org/10.1007/s00021-020-00529-z
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J. Neustupa. Measure-valued solutions of the Euler and Navier-Stokes equations for compressible barotropic fluids. Mathematische Nachrichten, 163(1):217–227, 1993. https://doi.org/10.1002/mana.19931630119
Y.-F. Yang, C.-S. Dou and Q.-C.Ju. Weak-strong uniqueness property for the magnetohydrodynamic equations of three-dimensional compressible isentropic flows. Nonlinear Analysis, 85(1):23–30, 2013. https://doi.org/10.1016/j.na.2013.02.015
J.-W. Zhang, S. Jiang and F. Xie. Global weak solutions of an initial boundary value problem for screw pinches in plasma physic. Mathematical Models and Methods in Applied Sciences, 19(06):833–875, 2009. https://doi.org/10.1142/S0218202509003644
S. Demoulini, D.M.A. Stuart and A.E. Tzavaras. Weak-strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics. Archive for Rational Mechanics and Analysis, 205:927–961, 2012. https://doi.org/10.1007/s00205-012-0523-6
R.J. DiPerna. Measure-valued solutions to conservation laws. Arch. Rational Mech. Anal., 88:223–270, 1985. https://doi.org/10.1007/BF00752112
B. Ducomet and E. Feireisl. The equations of magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars. Communications in Mathematical Physics, 266:595–629, 2006. https://doi.org/10.1007/s00220-006-0052-y
J.-S. Fan, S. Jiang and G. Nakamura. Weak-strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics. Communications in Mathematical Physics, 270:691–708, 2007.
E. Feireisl, P. Gwiazda, A. Świerczewska-Gwiazda and E. Wiedemann. Dissipative measure-valued solutions to the compressible Navier-Stokes system. Calculus of Variations and Partial Differential Equations, 55:141, 2016. https://doi.org/10.1007/s00526-016-1089-1
E. Feireisl, B.J. Jin and A. Novotný. Relative entropies, suitable weak solutions and weak strong uniqueness for the compressible Navier-Stokes system. Journal of Mathematical Fluid Mechanics, 14:717–730, 2012. https://doi.org/10.1007/s00021-011-0091-9
J.-C. Gao, Y.-H. Chen and Z.-A. Yao. Long-time behavior of solution to the compressible magnetohydrodynamic equations. Nonlinear Analysis, 128:122– 135, 2015. https://doi.org/10.1016/j.na.2015.07.028
C. He and Z.-P. Xin. On the regularity of weak solutions to the magnetohydrodynamic equations. Journal of Differential Equations, 213(2):235–254, 2005. https://doi.org/10.1016/j.jde.2004.07.002
G.-Y. Hong, X.-F. Hou, H.-Y. Peng and C.-J.Zhu. Global existence for a class of large solutions to three-dimensional compressible magnetohydrodynamic equations with vacuum. SIAM Journal on Mathematical Analysis, 49(4), 2017. https://doi.org/10.1137/16M1100447
X.-P. Hu and D.-H. Wang. Low Mach number limit of viscous compressible magnetohydrodynamic flow. SIAM Journal on Mathematical Analysis, 41(3):1272– 1294, 2009. https://doi.org/10.1137/080723983
X.-P. Hu and D.-H. Wang. Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows. Archive for Rational Mechanics and Analysis, 197:203–238, 2010. https://doi.org/10.1007/s00205-010-0295-9
B.-K. Huang. On the existence of dissipative measure-valued solutions to the compressible micropolar system. Journal of Mathematical Fluid Mechanics, 22(59), 2020. https://doi.org/10.1007/s00021-020-00529-z
S. Jiang, Q.-C. Ju and F.-C. Li. Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions. Communications in Mathematical Physics, 297:371–400, 2010. https://doi.org/10.1007/s00220-010-0992-0
J. Neustupa. Measure-valued solutions of the Euler and Navier-Stokes equations for compressible barotropic fluids. Mathematische Nachrichten, 163(1):217–227, 1993. https://doi.org/10.1002/mana.19931630119
Y.-F. Yang, C.-S. Dou and Q.-C.Ju. Weak-strong uniqueness property for the magnetohydrodynamic equations of three-dimensional compressible isentropic flows. Nonlinear Analysis, 85(1):23–30, 2013. https://doi.org/10.1016/j.na.2013.02.015
J.-W. Zhang, S. Jiang and F. Xie. Global weak solutions of an initial boundary value problem for screw pinches in plasma physic. Mathematical Models and Methods in Applied Sciences, 19(06):833–875, 2009. https://doi.org/10.1142/S0218202509003644