# On weak-strong uniqueness property for full compressible magnetohydrodynamics flows

Open Mathematics (2013)

- Volume: 11, Issue: 11, page 2005-2019
- ISSN: 2391-5455

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topWeiping Yan. "On weak-strong uniqueness property for full compressible magnetohydrodynamics flows." Open Mathematics 11.11 (2013): 2005-2019. <http://eudml.org/doc/268962>.

@article{WeipingYan2013,

abstract = {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.},

author = {Weiping Yan},

journal = {Open Mathematics},

keywords = {Magnetohydrodynamic flows; Weak solution; Strong solution; Entropy; magnetohydrodynamic flows; weak solution; strong solution; entropy},

language = {eng},

number = {11},

pages = {2005-2019},

title = {On weak-strong uniqueness property for full compressible magnetohydrodynamics flows},

url = {http://eudml.org/doc/268962},

volume = {11},

year = {2013},

}

TY - JOUR

AU - Weiping Yan

TI - On weak-strong uniqueness property for full compressible magnetohydrodynamics flows

JO - Open Mathematics

PY - 2013

VL - 11

IS - 11

SP - 2005

EP - 2019

AB - 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.

LA - eng

KW - Magnetohydrodynamic flows; Weak solution; Strong solution; Entropy; magnetohydrodynamic flows; weak solution; strong solution; entropy

UR - http://eudml.org/doc/268962

ER -

## References

top- [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 Zbl0984.35027
- [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 Zbl0448.73004
- [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 Zbl1113.76098
- [5] Eliezer S., Ghatak A., Hora H., An Introduction to Equations of State: Theory and Applications, Cambridge University Press, Cambridge-New York, 1986 Zbl1029.81002
- [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 Zbl1256.35054
- [8] Feireisl E., Novotný A., Singular Limits in Thermodynamics of Viscous Fluids, Adv. Math. Fluid Mech., Birkhäuser, Basel, 2009 Zbl1176.35126
- [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 Zbl1285.76034
- [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 Zbl1248.35143
- [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 Zbl1198.35179
- [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 Zbl1270.35342
- [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 Zbl1158.35075
- [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 Zbl1195.35253
- [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 Zbl1015.76071
- [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 Zbl1242.35051
- [17] Ladyzhenskaya O.A., The Mathematical Theory of Viscous Incompressible Flow, 2nd ed., Math. Appl., 2, Gordon and Breach, New York-London-Paris, 1969 Zbl0184.52603
- [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 Zbl1170.35500
- [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 Zbl0106.18302
- [22] Temam R., Navier-Stokes Equations. Theory and Numerical Analysis, Stud. Math. Appl., 2, North-Holland, Amsterdam-New York-Oxford, 1977 Zbl0383.35057

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