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doi: 10.3934/dcdsb.2021307
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## On weak (measure-valued)-strong uniqueness for compressible MHD system with non-monotone pressure law

 School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China

* Corresponding author: Ting Zhang

Received  June 2021 Revised  November 2021 Early access January 2022

In this paper, we define a renormalized dissipative measure-valued (rDMV) solution of the compressible magnetohydrodynamics (MHD) equations with non-monotone pressure law. We prove the existence of the rDMV solutions and establish a suitable relative energy inequality. And we obtain the weak (measure-valued)-strong uniqueness property of this rDMV solution with the help of the relative energy inequality.

Citation: Yu Liu, Ting Zhang. On weak (measure-valued)-strong uniqueness for compressible MHD system with non-monotone pressure law. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021307
##### References:
 [1] D. Bresch and P.-E. Jabin, Global existence of weak solutions for compressible Navier-Stokes equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor, Ann. of Math., 188 (2018), 577-684.  doi: 10.4007/annals.2018.188.2.4. [2] J. Březina, E. Feireisl and A. Novotný, Stability of strong solutions to the Navier-Stokes-Fourier system, SIAM J. Math. Anal., 52 (2020), 1761-1785.  doi: 10.1137/18M1223022. [3] H. Cabannes, Theoretical Magnetofluiddynamics, Academic Press, 1970. [4] T. Chang, B. J. Jin and A. Novotný, Compressible Navier-Stokes system with general inflow-out flow boundary data, SIAM J. Math. Anal., 51 (2019), 1238-1278.  doi: 10.1137/17M115089X. [5] N. Chaudhuri, On weak-strong uniqueness for compressible Navier-Stokes system with general pressure laws, Nonlinear Anal. Real World Appl., 49 (2019), 250-267.  doi: 10.1016/j.nonrwa.2019.03.004. [6] N. Chaudhuri, On weak (measure-valued)-strong uniqueness for compressible Navier-Stokes system with non-monotone pressure law, J. Math. Fluid Mech., 22 (2020), Paper No. 17, 13 pp. doi: 10.1007/s00021-019-0465-y. [7] G.-Q. Chen and D. Wang, Global solutions of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations, 182 (2002), 344-376.  doi: 10.1006/jdeq.2001.4111. [8] G.-Q. Chen and D. Wang, Existence and continuous dependence of large solutions for the magnetohydrodynamic equations, Z. Angew. Math. Phys., 54 (2003), 608-632.  doi: 10.1007/s00033-003-1017-z. [9] R. J. Diperna, Measure-valued solutions to conservation laws, Arch. Ration. Mech. Anal., 88 (1985), 223-270.  doi: 10.1007/BF00752112. [10] B. Ducomet and E. Feireisl, The equations of magnetohydrodynamics: on the interaction between matter and radiation in the evolution of gaseous stars, Comm. Math. Phys., 266 (2006), 595-629.  doi: 10.1007/s00220-006-0052-y. [11] J. Fan, S. Jiang and G. Nakamura, Vanishing shear viscosity limit in the magnetohydrodynamic equations, Comm. Math. Phys., 270 (2007), 691-708.  doi: 10.1007/s00220-006-0167-1. [12] D. Fang and T. Zhang, Compressible Navier-Stokes equations with vacuum state in the case of general pressure law, Math. Methods Appl. Sci., 29 (2006), 1081-1106.  doi: 10.1002/mma.708. [13] E. Feireisl, On compactness of solutions to the compressible isentropic Navier-Stokes equations when the density is not square integrable, Comment. Math. Univ. Carolin., 42 (2001), 83-98. [14] E. Feireisl, Compressible Navier-Stokes equations with a non-monotone pressure law, J. Differential Equations, 184 (2002), 97-108.  doi: 10.1006/jdeq.2001.4137. [15] E. Feireisl, On weak-strong uniqueness for the compressible Navier-Stokes system with non-monotone pressure law, Comm. Partial Differential Equations, 44 (2019), 271-278.  doi: 10.1080/03605302.2018.1543319. [16] E. Feireisl, P. Gwiazda, A. Świerczewska-Gwiazda and E. Wiedemann, Dissipative measure-valued solutions to the compressible Navier-Stokes system, Calc. Var. Partial Differential Equations, 55 (2016), Art. 141, 20 pp. doi: 10.1007/s00526-016-1089-1. [17] E. Feireisl, B. J. Jin and A. Novotný, Relative entropies, suitable weak solutions and weak-strong uniqueness for the compressible Navier-Stokes system, J. Math. Fluid Mech., 14 (2012), 717-730.  doi: 10.1007/s00021-011-0091-9. [18] E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392.  doi: 10.1007/PL00000976. [19] P. G. Fernández-Dalgo and O. Jarrín, Weak-strong uniqueness in weighted $L^2$ spaces and weak suitable solutions in local Morrey spaces for the MHD equations, J. Differential Equations, 271 (2021), 864-915.  doi: 10.1016/j.jde.2020.09.017. [20] H. Freistühler and P. Szmolyan, Existence and bifurcation of viscous profiles for all intermediate magnetohydrodynamic shock waves, SIAM J. Math. Anal., 26 (1995), 112-128.  doi: 10.1137/S0036141093247366. [21] P. Germain, Weak-strong uniqueness for the isentropic compressible Navier-Stokes system, J. Math. Fluid Mech., 13 (2011), 137-146.  doi: 10.1007/s00021-009-0006-1. [22] P. Gwiazda, A. Świerczewska-Gwiazda and E. Wiedemann, Weak-strong uniqueness for measure-valued solutions of some compressible fluid models, Nonlinearity, 28 (2015), 3873-3890.  doi: 10.1088/0951-7715/28/11/3873. [23] D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differential Equations, 120 (1995), 215-254.  doi: 10.1006/jdeq.1995.1111. [24] D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data, Arch. Ration. Mech. Anal., 132 (1995), 1-14.  doi: 10.1007/BF00390346. [25] D. Hoff and E. Tsyganov, Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics, Z. Angew. Math. Phys., 56 (2005), 791-804.  doi: 10.1007/s00033-005-4057-8. [26] X. Hu and D. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238.  doi: 10.1007/s00205-010-0295-9. [27] P.-L. Lions, Mathematical Topics in Fluid Mechanics, vol. 2. Compressible Models, New York: The Clarendon Press; Oxford University Press, 1998. [28] Y. Liu and T. Zhang, Global weak solutions to a 2D compressible non-resistivity MHD system with non-monotone pressure law and nonconstant viscosity, J. Math. Anal. Appl., 502 (2021), Paper No. 125244, 38 pp. doi: 10.1016/j.jmaa.2021.125244. [29] J. Neustupa, Measure-valued solutions of the Euler and Navier-Stokes equations for compressible barotropic fluids, Math. Nachr., 163 (1993), 217-227.  doi: 10.1002/mana.19931630119. [30] D. Wang, Large solutions to the initial-boundary value problem for planar magnetohydrodynamics, SIAM J. Appl. Math., 63 (2003), 1424-1441.  doi: 10.1137/S0036139902409284. [31] W. Yan, On weak-strong uniqueness property for full compressible magnetohydrodynamics flows, Cent. Eur. J. Math., 11 (2013), 2005-2019.  doi: 10.2478/s11533-013-0297-6. [32] Y.-F. Yang, C. Dou and Q. Ju, Weak-strong uniqueness property for the magnetohydrodynamic equations of three-dimensional compressible isentropic flows, Nonlinear Anal., 85 (2013), 23-30.  doi: 10.1016/j.na.2013.02.015.

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##### References:
 [1] D. Bresch and P.-E. Jabin, Global existence of weak solutions for compressible Navier-Stokes equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor, Ann. of Math., 188 (2018), 577-684.  doi: 10.4007/annals.2018.188.2.4. [2] J. Březina, E. Feireisl and A. Novotný, Stability of strong solutions to the Navier-Stokes-Fourier system, SIAM J. Math. Anal., 52 (2020), 1761-1785.  doi: 10.1137/18M1223022. [3] H. Cabannes, Theoretical Magnetofluiddynamics, Academic Press, 1970. [4] T. Chang, B. J. Jin and A. Novotný, Compressible Navier-Stokes system with general inflow-out flow boundary data, SIAM J. Math. Anal., 51 (2019), 1238-1278.  doi: 10.1137/17M115089X. [5] N. Chaudhuri, On weak-strong uniqueness for compressible Navier-Stokes system with general pressure laws, Nonlinear Anal. Real World Appl., 49 (2019), 250-267.  doi: 10.1016/j.nonrwa.2019.03.004. [6] N. Chaudhuri, On weak (measure-valued)-strong uniqueness for compressible Navier-Stokes system with non-monotone pressure law, J. Math. Fluid Mech., 22 (2020), Paper No. 17, 13 pp. doi: 10.1007/s00021-019-0465-y. [7] G.-Q. Chen and D. Wang, Global solutions of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations, 182 (2002), 344-376.  doi: 10.1006/jdeq.2001.4111. [8] G.-Q. Chen and D. Wang, Existence and continuous dependence of large solutions for the magnetohydrodynamic equations, Z. Angew. Math. Phys., 54 (2003), 608-632.  doi: 10.1007/s00033-003-1017-z. [9] R. J. Diperna, Measure-valued solutions to conservation laws, Arch. Ration. Mech. Anal., 88 (1985), 223-270.  doi: 10.1007/BF00752112. [10] B. Ducomet and E. Feireisl, The equations of magnetohydrodynamics: on the interaction between matter and radiation in the evolution of gaseous stars, Comm. Math. Phys., 266 (2006), 595-629.  doi: 10.1007/s00220-006-0052-y. [11] J. Fan, S. Jiang and G. Nakamura, Vanishing shear viscosity limit in the magnetohydrodynamic equations, Comm. Math. Phys., 270 (2007), 691-708.  doi: 10.1007/s00220-006-0167-1. [12] D. Fang and T. Zhang, Compressible Navier-Stokes equations with vacuum state in the case of general pressure law, Math. Methods Appl. Sci., 29 (2006), 1081-1106.  doi: 10.1002/mma.708. [13] E. Feireisl, On compactness of solutions to the compressible isentropic Navier-Stokes equations when the density is not square integrable, Comment. Math. Univ. Carolin., 42 (2001), 83-98. [14] E. Feireisl, Compressible Navier-Stokes equations with a non-monotone pressure law, J. Differential Equations, 184 (2002), 97-108.  doi: 10.1006/jdeq.2001.4137. [15] E. Feireisl, On weak-strong uniqueness for the compressible Navier-Stokes system with non-monotone pressure law, Comm. Partial Differential Equations, 44 (2019), 271-278.  doi: 10.1080/03605302.2018.1543319. [16] E. Feireisl, P. Gwiazda, A. Świerczewska-Gwiazda and E. Wiedemann, Dissipative measure-valued solutions to the compressible Navier-Stokes system, Calc. Var. Partial Differential Equations, 55 (2016), Art. 141, 20 pp. doi: 10.1007/s00526-016-1089-1. [17] E. Feireisl, B. J. Jin and A. Novotný, Relative entropies, suitable weak solutions and weak-strong uniqueness for the compressible Navier-Stokes system, J. Math. Fluid Mech., 14 (2012), 717-730.  doi: 10.1007/s00021-011-0091-9. [18] E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392.  doi: 10.1007/PL00000976. [19] P. G. Fernández-Dalgo and O. Jarrín, Weak-strong uniqueness in weighted $L^2$ spaces and weak suitable solutions in local Morrey spaces for the MHD equations, J. Differential Equations, 271 (2021), 864-915.  doi: 10.1016/j.jde.2020.09.017. [20] H. Freistühler and P. Szmolyan, Existence and bifurcation of viscous profiles for all intermediate magnetohydrodynamic shock waves, SIAM J. Math. Anal., 26 (1995), 112-128.  doi: 10.1137/S0036141093247366. [21] P. Germain, Weak-strong uniqueness for the isentropic compressible Navier-Stokes system, J. Math. Fluid Mech., 13 (2011), 137-146.  doi: 10.1007/s00021-009-0006-1. [22] P. Gwiazda, A. Świerczewska-Gwiazda and E. Wiedemann, Weak-strong uniqueness for measure-valued solutions of some compressible fluid models, Nonlinearity, 28 (2015), 3873-3890.  doi: 10.1088/0951-7715/28/11/3873. [23] D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differential Equations, 120 (1995), 215-254.  doi: 10.1006/jdeq.1995.1111. [24] D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data, Arch. Ration. Mech. Anal., 132 (1995), 1-14.  doi: 10.1007/BF00390346. [25] D. Hoff and E. Tsyganov, Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics, Z. Angew. Math. Phys., 56 (2005), 791-804.  doi: 10.1007/s00033-005-4057-8. [26] X. Hu and D. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238.  doi: 10.1007/s00205-010-0295-9. [27] P.-L. Lions, Mathematical Topics in Fluid Mechanics, vol. 2. Compressible Models, New York: The Clarendon Press; Oxford University Press, 1998. [28] Y. Liu and T. Zhang, Global weak solutions to a 2D compressible non-resistivity MHD system with non-monotone pressure law and nonconstant viscosity, J. Math. Anal. Appl., 502 (2021), Paper No. 125244, 38 pp. doi: 10.1016/j.jmaa.2021.125244. [29] J. Neustupa, Measure-valued solutions of the Euler and Navier-Stokes equations for compressible barotropic fluids, Math. Nachr., 163 (1993), 217-227.  doi: 10.1002/mana.19931630119. [30] D. Wang, Large solutions to the initial-boundary value problem for planar magnetohydrodynamics, SIAM J. Appl. Math., 63 (2003), 1424-1441.  doi: 10.1137/S0036139902409284. [31] W. Yan, On weak-strong uniqueness property for full compressible magnetohydrodynamics flows, Cent. Eur. J. Math., 11 (2013), 2005-2019.  doi: 10.2478/s11533-013-0297-6. [32] Y.-F. Yang, C. Dou and Q. Ju, Weak-strong uniqueness property for the magnetohydrodynamic equations of three-dimensional compressible isentropic flows, Nonlinear Anal., 85 (2013), 23-30.  doi: 10.1016/j.na.2013.02.015.
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