February  2022, 27(2): 1001-1027. doi: 10.3934/dcdsb.2021078

Energy equality for weak solutions to the 3D magnetohydrodynamic equations in a bounded domain

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 Published  February 2022 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.

Citation: Guodong Wang, Bijun Zuo. Energy equality for weak solutions to the 3D magnetohydrodynamic equations in a bounded domain. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 1001-1027. doi: 10.3934/dcdsb.2021078
References:
[1]

I. AkramovT. DebiecJ. 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.

[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.

[3]

C. BardosE. 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.

[4]

R. E. CaflischI. 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.

[5]

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

[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.

[7]

A. CheskidovP. ConstantinS. 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.

[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.

[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.

[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.

[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.

[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.

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[14]

E. FeireislP. GwiazdaA. 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.

[15]

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

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[23]

Q.-H. NguyenP.-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.

[24]

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

[25]

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

[26]

H. PolitanoA. 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.

[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.

[28]

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

[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.

[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.

[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.

[32]

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

[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.

[34]

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

[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.

[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.

show all references

References:
[1]

I. AkramovT. DebiecJ. 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.

[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.

[3]

C. BardosE. 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.

[4]

R. E. CaflischI. 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.

[5]

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

[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.

[7]

A. CheskidovP. ConstantinS. 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.

[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.

[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.

[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.

[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.

[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.

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

E. FeireislP. GwiazdaA. 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.

[15]

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

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[23]

Q.-H. NguyenP.-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.

[24]

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

[25]

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

[26]

H. PolitanoA. 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.

[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.

[28]

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

[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.

[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.

[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.

[32]

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

[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.

[34]

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

[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.

[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.

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