This paper is devoted to the study of the weak-strong uniqueness property for full compressible magnetohydrodynamics flows. The governing equations for magnetohydrodynamic flows are expressed by the full Navier-Stokes system for compressible fluids enhanced by forces due to the presence of the magnetic field as well as the gravity and an additional equation which describes the evolution of the magnetic field. Using the relative entropy inequality, we prove that a weak solution coincides with the strong solution, emanating from the same initial data, as long as the latter exists.
@article{bwmeta1.element.doi-10_2478_s11533-013-0297-6,
author = {Weiping Yan},
title = {On weak-strong uniqueness property for full compressible magnetohydrodynamics flows},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {2005-2019},
zbl = {1292.76079},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0297-6}
}
Weiping Yan. On weak-strong uniqueness property for full compressible magnetohydrodynamics flows. Open Mathematics, Tome 11 (2013) pp. 2005-2019. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0297-6/
[1] Carrillo J., Jüngel A., Markowich P.A., Toscani G., Unterreiter A., Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., 2001, 133(1), 1–82 http://dx.doi.org/10.1007/s006050170032 | Zbl 0984.35027
[2] Chemin J.-Y., About weak-strong uniqueness for the 3D incompressible Navier-Stokes system, Comm. Pure. Appl. Math., 2011, 64(12), 1587–1598 http://dx.doi.org/10.1002/cpa.20386
[3] Dafermos C.M., The second law of thermodynamics and stability, Arch. Rational Mech. Anal., 1979, 70(2), 167–179 http://dx.doi.org/10.1007/BF00250353 | Zbl 0448.73004
[4] Ducomet B., Feireisl E., The equations of magnetohydrodynamics: on the interaction between matter and radiation in the evolution of gaseous stars, Comm. Math. Phys., 2006, 266(3), 595–629 http://dx.doi.org/10.1007/s00220-006-0052-y | Zbl 1113.76098
[5] Eliezer S., Ghatak A., Hora H., An Introduction to Equations of State: Theory and Applications, Cambridge University Press, Cambridge-New York, 1986 | Zbl 1029.81002
[6] Feireisl E., Dynamics of Viscous Compressible Fluids, Oxford Lecture Ser. Math. Appl., 26, Oxford University Press, Oxford, 2004
[7] Feireisl E., Jin B.J., Novotný A., Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system, J. Math. Fluid Mech., 2012, 14(4), 717–730 http://dx.doi.org/10.1007/s00021-011-0091-9 | Zbl 1256.35054
[8] Feireisl E., Novotný A., Singular Limits in Thermodynamics of Viscous Fluids, Adv. Math. Fluid Mech., Birkhäuser, Basel, 2009 | Zbl 1176.35126
[9] Feireisl E., Novotný A., Weak-strong uniqueness property for the full Navier-Stokes-Fourier system, Arch. Ration. Mech. Anal., 2012, 204(2), 683–706 http://dx.doi.org/10.1007/s00205-011-0490-3 | Zbl 1285.76034
[10] Feireisl E., Novotný A., Sun Y., Suitable weak solutions to the Navier-Stokes equations of compressible viscous fluids, Indiana Univ. Math. J., 2011, 60(2), 611–631 http://dx.doi.org/10.1512/iumj.2011.60.4406 | Zbl 1248.35143
[11] Germain P., Strong solutions and weak-strong uniqueness for the nonhomogeneous Navier-Stokes system, J. Anal. Math., 2008, 105, 169–196 http://dx.doi.org/10.1007/s11854-008-0034-4 | Zbl 1198.35179
[12] Germain P., Weak-strong uniqueness for the isentropic compressible Navier-Stokes system, J. Math. Fluid Mech., 2011, 13(1), 169–196 http://dx.doi.org/10.1007/s00021-009-0006-1 | Zbl 1270.35342
[13] Hu X., Wang D., Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys., 2008, 283(1), 255–284 http://dx.doi.org/10.1007/s00220-008-0497-2 | Zbl 1158.35075
[14] Jiang S., Ju Q., Li F., Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions, Comm. Math. Phys., 2010, 297(2), 371–400 http://dx.doi.org/10.1007/s00220-010-0992-0 | Zbl 1195.35253
[15] 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., 2001, 39(1-4), 261–343 http://dx.doi.org/10.1023/A:1004844002437 | Zbl 1015.76071
[16] Kwon Y.-S., Trivisa K., On the incompressible limits for the full magnetohydrodynamics flows, J. Differential Equations, 2011, 251(7), 1990–2023 http://dx.doi.org/10.1016/j.jde.2011.04.016 | Zbl 1242.35051
[17] Ladyzhenskaya O.A., The Mathematical Theory of Viscous Incompressible Flow, 2nd ed., Math. Appl., 2, Gordon and Breach, New York-London-Paris, 1969 | Zbl 0184.52603
[18] Leray J., Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math., 1934, 63(1), 193–248 http://dx.doi.org/10.1007/BF02547354
[19] Lions P.-L., Mathematical Topics in Fluid Mechanics II. Compressible Models, Oxford Lecture Ser. Math. Appl., 10, Oxford Science Publication, Oxford University Press, Oxford-New York, 1998
[20] Saint-Raymond L., Hydrodynamic limits: some improvements of the relative entropy method, Ann. Inst. H. Poincarè Anal. Non Linéaire, 2009, 26(3), 705–744 http://dx.doi.org/10.1016/j.anihpc.2008.01.001 | Zbl 1170.35500
[21] Serrin J., On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 1962, 9, 187–195 http://dx.doi.org/10.1007/BF00253344 | Zbl 0106.18302
[22] Temam R., Navier-Stokes Equations. Theory and Numerical Analysis, Stud. Math. Appl., 2, North-Holland, Amsterdam-New York-Oxford, 1977 | Zbl 0383.35057