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doi: 10.3934/dcdsb.2021078
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Energy equality for weak solutions to the 3D magnetohydrodynamic equations in a bounded domain

Guodong Wang 1, and  Bijun Zuo 2,,

1. 

Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, China
2. 

College of Mathematical Sciences, Harbin Engineering University, Harbin 150001, China

* Corresponding author: Bijun Zuo

Received  September 2020 Revised  December 2020 Early access March 2021

In this paper, we study the energy equality for weak solutions to the 3D homogeneous incompressible magnetohydrodynamic equations with viscosity and magnetic diffusion in a bounded domain. Two types of regularity conditions are imposed on weak solutions to ensure the energy equality. For the first type, some global integrability condition for the velocity $ \mathbf u $ is required, while for the magnetic field $ \mathbf b $ and the magnetic pressure $ \pi $, some suitable integrability conditions near the boundary are sufficient. In contrast with the first type, the second type claims that if some additional interior integrability is imposed on $ \mathbf b $, then the regularity on $ \mathbf u $ can be relaxed.

Keywords: MHD equations, energy equality, weak solution, bounded domain.
Mathematics Subject Classification: 76W05, 35D30, 35Q35.
Citation: Guodong Wang, Bijun Zuo. Energy equality for weak solutions to the 3D magnetohydrodynamic equations in a bounded domain. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021078
References:
[1]

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C. Bardos, E. S. Titi and E. Wiedemann, Onsager's conjecture with physical boundaries and an application to the vanishing viscosity limit, Comm. Math. Phys., 370 (2019), 291-310.  doi: 10.1007/s00220-019-03493-6.  Google Scholar

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M. Chen, Z. Liang, D. Wang and R. Xu, Energy equality in compressible fluids with physical boundaries, SIAM J. Math. Anal., 52 (2020), 1363-1385.  doi: 10.1137/19M1287213.  Google Scholar

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R. M. Chen and C. Yu, Onsager's energy conservation for inhomogeneous Euler equations, J. Math. Pures Appl., 131 (2019), 1-16.  doi: 10.1016/j.matpur.2019.02.003.  Google Scholar

[7]

A. Cheskidov, P. Constantin, S. Friedlander and R. Shvydkoy, Energy conservation and Onsager's conjecture for the Euler equations, Nonlinearity, 21 (2008), 1233-1252.  doi: 10.1088/0951-7715/21/6/005.  Google Scholar

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T. D. Drivas and H. Q. Nguyen, Onsager's conjecture and anomalous dissipation on domains with doundary, SIAM J. Math. Anal., 50 (2018), 4785-4811.  doi: 10.1137/18M1178864.  Google Scholar

[10]

J. Duchon and R. Robert, Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations, Nonlinearity, 13 (2000), 249-255.  doi: 10.1088/0951-7715/13/1/312.  Google Scholar

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L. Escauriaza and S. Montaner, Some remarks on the $L^p$ regularity of second derivatives of solutions to non-divergence elliptic equations and the Dini condition, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 28 (2017), 49-63.  doi: 10.4171/RLM/751.  Google Scholar

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E. Feireisl, P. Gwiazda, A. Świerczewska-Gwiazda and E. Wiedemann, Regularity and energy conservation for the compressible Euler equations, Arch. Ration. Mech. Anal., 223 (2017), 1375-1395.  doi: 10.1007/s00205-016-1060-5.  Google Scholar

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E. Kang and J. Lee, Remarks on the magnetic helicity and energy conservation for ideal magneto-hydrodynamics, Nonlinearity, 20 (2007), 2681-2689.  doi: 10.1088/0951-7715/20/11/011.  Google Scholar

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I. Lacroix-Violet and A. Vasseur, Global weak solutions to the compressible quantum Navier-Stokes equation and its semi-classical limit, J. Math. Pures Appl., 114 (2018), 191-210.  doi: 10.1016/j.matpur.2017.12.002.  Google Scholar

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T. M. Leslie and R. Shvydkoy, The energy balance relation for weak solutions of the density-dependent Navier-Stokes equations, J. Differential Equations, 261 (2016), 3719-3733.  doi: 10.1016/j.jde.2016.06.001.  Google Scholar

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P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1. Incompressible Models. Oxford Lecture Series in Mathematics and its Applications, vol. 3. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1996.  Google Scholar

[22]

Q.-H. Nguyen and P.-T. Nguyen, Onsager's conjecture on the energy conservation for solutions of Euler equations in bounded domains, J. Nonlinear Sci., 29 (2019), 207-213.  doi: 10.1007/s00332-018-9483-9.  Google Scholar

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Q.-H. Nguyen, P.-T. Nguyen and B. Q. Tang, Energy conservation for inhomogeneous incompressible and compressible Euler equations, J. Differential Equations, 269 (2020), 7171-7210.  doi: 10.1016/j.jde.2020.05.025.  Google Scholar

[24]

Q.-H. Nguyen, P.-T. Nguyen and B. Q. Tang, Energy equalities for compressible Navier-Stokes equations, Nonlinearity, 32 (2019), 4206-4231.  doi: 10.1088/1361-6544/ab28ae.  Google Scholar

[25]

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[26]

H. Politano, A. Pouquet and P.-L. Sulem, Current and votticity dynamics in three-dimensional magnetohydrodynamics turbulence, Phys. Plasmas, 2 (1995), 2931-2939.  doi: 10.1063/1.871473.  Google Scholar

[27]

J. Serrin, The initial value problem for the Navier-Stokes equations. Nonlinear Problems. Proceedings of the Symposium, Madison, Wisconsin, 1962. University of Wisconsin Press, Madison, Wisconsin, 69-98, 1963.  Google Scholar

[28]

M. Shinbrot, The energy equation for the Navier-Stokes system, SIAM J. Math. Anal., 5 (1974), 948-954.  doi: 10.1137/0505092.  Google Scholar

[29]

J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[30]

T. Wang, X. Zhao, Y. Chen and M. Zhang, Energy conservation for the weak solutions to the equations of compressible magnetohydrodynamic flows in three dimensions, J. Math. Anal. Appl., 480 (2019), 123373, 18 pp. doi: 10.1016/j.jmaa.2019.07.063.  Google Scholar

[31]

Y. Wang and B. Zuo, Energy and cross-helicity conservation for the three-dimensional ideal MHD equations in a bounded domain, J. Differential Equations, 268 (2020), 4079-4101.  doi: 10.1016/j.jde.2019.10.045.  Google Scholar

[32]

C. Yu, A new proof to the energy conservation for the Navier-Stokes equations, arXiv: 1604.05697. Google Scholar

[33]

C. Yu, Energy conservation for the weak solutions of the compressible Navier-Stokes equations, Arch. Ration. Mech. Anal., 225 (2017), 1073-1087.  doi: 10.1007/s00205-017-1121-4.  Google Scholar

[34]

C. Yu, The energy equality for the Navier-Stokes equations in bounded domains, arXiv: 1802.07661. Google Scholar

[35]

X. Yu, A note on the energy conservation of the ideal MHD equations, Nonlinearity, 22 (2009), 913-922.  doi: 10.1088/0951-7715/22/4/012.  Google Scholar

[36]

Y. Zhou, Remarks on regularities for the 3D MHD equations, Discrete Contin. Dyn. Syst., 12 (2005), 881-886.  doi: 10.3934/dcds.2005.12.881.  Google Scholar

show all references

References:
[1]

I. Akramov, T. Debiec, J. Skipper and E. Wiedemann, Energy conservation for the compressible Euler and Navier-Stokes equations with vacuum, Anal. PDE, 13 (2020), 789-811.  doi: 10.2140/apde.2020.13.789.  Google Scholar

[2]

C. Bardos and E. S. Titi, Onsager's conjecture for the incompressible Euler equations in bounded domains, Arch. Ration. Mech. Anal., 228 (2018), 197-207.  doi: 10.1007/s00205-017-1189-x.  Google Scholar

[3]

C. Bardos, E. S. Titi and E. Wiedemann, Onsager's conjecture with physical boundaries and an application to the vanishing viscosity limit, Comm. Math. Phys., 370 (2019), 291-310.  doi: 10.1007/s00220-019-03493-6.  Google Scholar

[4]

R. E. Caflisch, I. Klapper and G. Steele, Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD, Comm. Math. Phys., 184 (1997), 443-455.  doi: 10.1007/s002200050067.  Google Scholar

[5]

M. Chen, Z. Liang, D. Wang and R. Xu, Energy equality in compressible fluids with physical boundaries, SIAM J. Math. Anal., 52 (2020), 1363-1385.  doi: 10.1137/19M1287213.  Google Scholar

[6]

R. M. Chen and C. Yu, Onsager's energy conservation for inhomogeneous Euler equations, J. Math. Pures Appl., 131 (2019), 1-16.  doi: 10.1016/j.matpur.2019.02.003.  Google Scholar

[7]

A. Cheskidov, P. Constantin, S. Friedlander and R. Shvydkoy, Energy conservation and Onsager's conjecture for the Euler equations, Nonlinearity, 21 (2008), 1233-1252.  doi: 10.1088/0951-7715/21/6/005.  Google Scholar

[8]

P. Constantin and W. E and E. S. Titi, Onsager's conjecture on the energy conservation for solutions of Euler's equation, Comm. Math. Phys., 165 (1994), 207-209. doi: 10.1007/BF02099744.  Google Scholar

[9]

T. D. Drivas and H. Q. Nguyen, Onsager's conjecture and anomalous dissipation on domains with doundary, SIAM J. Math. Anal., 50 (2018), 4785-4811.  doi: 10.1137/18M1178864.  Google Scholar

[10]

J. Duchon and R. Robert, Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations, Nonlinearity, 13 (2000), 249-255.  doi: 10.1088/0951-7715/13/1/312.  Google Scholar

[11]

L. Escauriaza and S. Montaner, Some remarks on the $L^p$ regularity of second derivatives of solutions to non-divergence elliptic equations and the Dini condition, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 28 (2017), 49-63.  doi: 10.4171/RLM/751.  Google Scholar

[12]

G. L. Eyink, Energy dissipation without viscosity in ideal hydrodynamics, I. Fourier analysis and local energy transfer, Phys. D, 78 (1994), 222-240.  doi: 10.1016/0167-2789(94)90117-1.  Google Scholar

[13] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford Lecture Series in Mathematics and its Applications, vol. 26. Oxford University Press, Oxford, 2004.   Google Scholar
[14]

E. Feireisl, P. Gwiazda, A. Świerczewska-Gwiazda and E. Wiedemann, Regularity and energy conservation for the compressible Euler equations, Arch. Ration. Mech. Anal., 223 (2017), 1375-1395.  doi: 10.1007/s00205-016-1060-5.  Google Scholar

[15]

A. Hasegawa, Self-organization processes in continous media, Adv. in Physics, 34 (1985), 1-42.  doi: 10.1080/00018738500101721.  Google Scholar

[16]

C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254.  doi: 10.1016/j.jde.2004.07.002.  Google Scholar

[17]

E. Kang and J. Lee, Remarks on the magnetic helicity and energy conservation for ideal magneto-hydrodynamics, Nonlinearity, 20 (2007), 2681-2689.  doi: 10.1088/0951-7715/20/11/011.  Google Scholar

[18]

A. Kufner, O. John and S. Fu${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over c} }}$ík, Function Spaces, Academia, Prague, 1977.  Google Scholar

[19]

I. Lacroix-Violet and A. Vasseur, Global weak solutions to the compressible quantum Navier-Stokes equation and its semi-classical limit, J. Math. Pures Appl., 114 (2018), 191-210.  doi: 10.1016/j.matpur.2017.12.002.  Google Scholar

[20]

T. M. Leslie and R. Shvydkoy, The energy balance relation for weak solutions of the density-dependent Navier-Stokes equations, J. Differential Equations, 261 (2016), 3719-3733.  doi: 10.1016/j.jde.2016.06.001.  Google Scholar

[21]

P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1. Incompressible Models. Oxford Lecture Series in Mathematics and its Applications, vol. 3. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1996.  Google Scholar

[22]

Q.-H. Nguyen and P.-T. Nguyen, Onsager's conjecture on the energy conservation for solutions of Euler equations in bounded domains, J. Nonlinear Sci., 29 (2019), 207-213.  doi: 10.1007/s00332-018-9483-9.  Google Scholar

[23]

Q.-H. Nguyen, P.-T. Nguyen and B. Q. Tang, Energy conservation for inhomogeneous incompressible and compressible Euler equations, J. Differential Equations, 269 (2020), 7171-7210.  doi: 10.1016/j.jde.2020.05.025.  Google Scholar

[24]

Q.-H. Nguyen, P.-T. Nguyen and B. Q. Tang, Energy equalities for compressible Navier-Stokes equations, Nonlinearity, 32 (2019), 4206-4231.  doi: 10.1088/1361-6544/ab28ae.  Google Scholar

[25]

L. Onsager, Statistical hydrodynamics, Nuovo Cimento (9) 6, (Supplemento, 2 (Convegno Internazionale di Meccanica Statistica)), (1949), 279–287. doi: 10.1007/BF02780991.  Google Scholar

[26]

H. Politano, A. Pouquet and P.-L. Sulem, Current and votticity dynamics in three-dimensional magnetohydrodynamics turbulence, Phys. Plasmas, 2 (1995), 2931-2939.  doi: 10.1063/1.871473.  Google Scholar

[27]

J. Serrin, The initial value problem for the Navier-Stokes equations. Nonlinear Problems. Proceedings of the Symposium, Madison, Wisconsin, 1962. University of Wisconsin Press, Madison, Wisconsin, 69-98, 1963.  Google Scholar

[28]

M. Shinbrot, The energy equation for the Navier-Stokes system, SIAM J. Math. Anal., 5 (1974), 948-954.  doi: 10.1137/0505092.  Google Scholar

[29]

J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[30]

T. Wang, X. Zhao, Y. Chen and M. Zhang, Energy conservation for the weak solutions to the equations of compressible magnetohydrodynamic flows in three dimensions, J. Math. Anal. Appl., 480 (2019), 123373, 18 pp. doi: 10.1016/j.jmaa.2019.07.063.  Google Scholar

[31]

Y. Wang and B. Zuo, Energy and cross-helicity conservation for the three-dimensional ideal MHD equations in a bounded domain, J. Differential Equations, 268 (2020), 4079-4101.  doi: 10.1016/j.jde.2019.10.045.  Google Scholar

[32]

C. Yu, A new proof to the energy conservation for the Navier-Stokes equations, arXiv: 1604.05697. Google Scholar

[33]

C. Yu, Energy conservation for the weak solutions of the compressible Navier-Stokes equations, Arch. Ration. Mech. Anal., 225 (2017), 1073-1087.  doi: 10.1007/s00205-017-1121-4.  Google Scholar

[34]

C. Yu, The energy equality for the Navier-Stokes equations in bounded domains, arXiv: 1802.07661. Google Scholar

[35]

X. Yu, A note on the energy conservation of the ideal MHD equations, Nonlinearity, 22 (2009), 913-922.  doi: 10.1088/0951-7715/22/4/012.  Google Scholar

[36]

Y. Zhou, Remarks on regularities for the 3D MHD equations, Discrete Contin. Dyn. Syst., 12 (2005), 881-886.  doi: 10.3934/dcds.2005.12.881.  Google Scholar

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