November  2022, 42(11): 5487-5508. doi: 10.3934/dcds.2022110

Energy conservation and regularity for the 3D magneto-hydrodynamics equations

Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, China

*Corresponding author: Fan Wu

Received  July 2021 Revised  June 2022 Published  November 2022 Early access  August 2022

Fund Project: The authors are supported by the Construct Program of the Key Discipline in Hunan Province and NSFC (Grant No. 11871209), and the Hunan Provincial NSF (No. 2022JJ10033)

This paper studies the energy conservation and regularity problems for the 3D magneto-hydrodynamics (MHD) equations. We first establish some uniform bounds on some invariant quantities in terms of suitable weak solution $ (u,b)\in L^{2,\infty}(0,T;BMO(\Omega)) $. As the applications, first, we show that as the solution $ (u,b) $ approaches a finite blowup time $ T $, the $ BMO $ norm must blow up at a rate $ \frac{c}{\sqrt{T-t}} $ with some absolute constant $ c>0 $. Then, a regularity criteria for suitable weak solutions is proved which allows the vertical part of the velocity and magnetic to be large under the norm of $ L^{2,\infty}\left([-1,0]; BMO(\mathbb{R}^3)\right) $. Finally, we prove that any suitable weak solution of the MHD equations in $ L^{2,\infty}(0, T; BMO (\Omega)) $ satisfies the local energy equality for any bounded Lipschitz domain $ \Omega\subseteq\mathbb{R}^3 $. As a corollary, we prove that any suitable weak solution of MHD equations in $ L^{2,\infty}(0, T; BMO_{loc} (\mathbb{R}^3)) $ satisfies the energy equality.

Citation: Wenke Tan, Fan Wu. Energy conservation and regularity for the 3D magneto-hydrodynamics equations. Discrete and Continuous Dynamical Systems, 2022, 42 (11) : 5487-5508. doi: 10.3934/dcds.2022110
References:
[1]

H. Beirão Da Veiga, A new regularity class for the Navier-Stokes equations in $\mathbb{R}^n$, Chinese Ann. Math. Ser. B, 16 (1995), 407-412. 

[2]

L. Berselli and E. Chiodaroli, On the energy equality for the 3D Navier-Stokes equations, Nonlinear Analysis, 192 (2020), 111704.

[3]

J. Bourgain and N. Pavlović, Ill-posedness of the Navier-Stokes equations in a critical space in 3D, Journal of Functional Analysis, 255 (2008), 2233-2247. 

[4]

T. Buckmaster and V. Vicol, Nonuniqueness of weak solutions to the Navier-Stokes equation, Annals of Mathematics, 189 (2019), 101-144.  doi: 10.4007/annals.2019.189.1.3.

[5]

H. Cabannes, Theoretical Magnetofluiddynamics, Academic Press, New York and London, 1970.

[6]

R. E. CaflischI. Klapper and G. Steele, Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD, Communications in Mathematical Physics, 184 (1997), 443-455.  doi: 10.1007/s002200050067.

[7]

M. Cannone, Ondelettes, Paraproduits et Navier-Stokes, Paris 9, 1995.

[8]

C. Cao and E. S. Titi, Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor, Archive for Rational Mechanics and Analysis, 202 (2011), 919-932.  doi: 10.1007/s00205-011-0439-6.

[9]

C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, Journal of Differential Equations, 248 (2010), 2263-2274.  doi: 10.1016/j.jde.2009.09.020.

[10]

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

[11]

A. Cheskidov, S. Friedlander and R. Shvydkoy, On the energy equality for weak solutions of the 3D Navier-Stokes equations, Advances in Mathematical Fluid Mechanics. Springer, Berlin, Heidelberg, (2010), 171–175. doi: 10.1007/978-3-642-04068-9_10.

[12]

A. Cheskidov and X. Luo, Energy equality for the Navier-Stokes equations in weak-in-time Onsager spaces, Nonlinearity, 33 (2020), 1388-1403.  doi: 10.1088/1361-6544/ab60d3.

[13]

P. ConstantinW. E and E. S. Titi, Onsager's conjecture on the energy conservation for solutions of Euler's equation, Communications in Mathematical Physics, 165 (1994), 207-209.  doi: 10.1007/BF02099744.

[14]

G. Duvaut and J.-L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Archive for Rational Mechanics and Analysis, 46 (1972), 241-279.  doi: 10.1007/BF00250512.

[15]

L. EscauriazaG. Serë gin and V. Shverak, $L_{3, \infty}$-solutions of the Navier-Stokes equations and backward uniqueness, Russian Mathematical Surveys, 58 (2003), 211-250.  doi: 10.1070/RM2003v058n02ABEH000609.

[16]

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

[17]

C. He and Y. Wang, Limiting case for the regularity criterion of the Navier-Stokes equations and the magnetohydrodynamic equations, Science China Mathematics, 53 (2010), 1767-1774.  doi: 10.1007/s11425-010-3135-3.

[18]

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

[19]

C. He and Z. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, Journal of Functional Analysis, 227 (2005), 113-152.  doi: 10.1016/j.jfa.2005.06.009.

[20]

P. Isett, A proof of Onsager's conjecture, Annals of Mathematics, 188 (2018), 871-963.  doi: 10.4007/annals.2018.188.3.4.

[21]

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.

[22]

H. Kim and H. Kozono, Interior regularity criteria in weak spaces for the Navier-Stokes equations, Manuscripta Mathematica, 115 (2004), 85-100.  doi: 10.1007/s00229-004-0484-7.

[23]

J.-M. Kim, The energy conservations and lower bounds for possible singular solutions to the 3D incompressible MHD equations, Acta Math. Sci. Ser. B (Engl. Ed.), 40 (2020), 237-244.  doi: 10.1007/s10473-020-0116-x.

[24]

H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Advances in Mathematics, 157 (2001), 22-35. 

[25]

H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations, Math. Z., 235 (2000), 173-194.  doi: 10.1007/s002090000130.

[26]

I. Kukavica, Role of the pressure for validity of the energy equality for solutions of the Navier-Stokes equation, Journal of Dynamics and Differential Equations, 18 (2006), 461-482.  doi: 10.1007/s10884-006-9010-9.

[27]

T. M. Leslie and R. Shvydkoy, The energy measure for the Euler and Navier-Stokes equations, Archive for Rational Mechanics and Analysis, 230 (2018), 459-492.  doi: 10.1007/s00205-018-1250-4.

[28]

T. M. Leslie and R. Shvydkoy, Conditions implying energy equality for weak solutions of the Navier-Stokes equations, SIAM Journal on Mathematical Analysis, 50 (2018), 870-890.  doi: 10.1137/16M1104147.

[29]

J. L. Lions, Sur la régularité et l'unicité des solutions turbulentes des équations de Navier-Stokes, Rendiconti del Seminario Matematico della Universita di Padova, 30 (1960), 16-23. 

[30]

E. Miller, A regularity criterion for the Navier-Stokes equation involving only the middle eigenvalue of the strain tensor, Archive for Rational Mechanics and Analysis, 235 (2020), 99-139.  doi: 10.1007/s00205-019-01419-z.

[31]

L. Onsager, Statistical Hydrodynamics, Nuovo Cimento (Supplemento), 6 (1949), 279-287.  doi: 10.1007/BF02780991.

[32]

G. Prodi, Un teorema di unicita per le equazioni di Navier-Stokes, Annali di Matematica pura ed Applicata, 48 (1959), 173-182.  doi: 10.1007/BF02410664.

[33]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Communications on Pure and Applied Mathematics, 36 (1983), 635-664.  doi: 10.1002/cpa.3160360506.

[34]

J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Archive for Rational Mechanics and Analysis, 9 (1962), 187-195.  doi: 10.1007/BF00253344.

[35]

M. Shinbrot, The energy equation for the Navier-Stokes system, SIAM Journal on Mathematical Analysis, 5 (1974), 948-954.  doi: 10.1137/0505092.

[36]

W. Tan and Z. Yin, The energy measure for the Navier-Stokes and Euler equations and some applications, Submitted.

[37]

B. Wang, Ill-posedness for the Navier-Stokes equations in critical Besov spaces $\dot{B}_{\infty, q}^{-1}$, Advances in Mathematics, 268 (2015), 350-372.  doi: 10.1016/j.aim.2014.09.024.

[38]

W. Wang and Z. Zhang, On the interior regularity criteria and the number of singular points to the Navier-Stokes equations, Journal d'Analyse Mathematique, 123 (2014), 139-170.  doi: 10.1007/s11854-014-0016-7.

[39]

Y. Yong and Q. S. Jiu, Energy equality and uniqueness of weak solutions to MHD equations in $L^\infty(0, T; L^n (\Omega))$, Acta Mathematica Sinica, English Series, 25 (2009), 803-814.  doi: 10.1007/s10114-009-7214-8.

[40]

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

[41]

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.

show all references

References:
[1]

H. Beirão Da Veiga, A new regularity class for the Navier-Stokes equations in $\mathbb{R}^n$, Chinese Ann. Math. Ser. B, 16 (1995), 407-412. 

[2]

L. Berselli and E. Chiodaroli, On the energy equality for the 3D Navier-Stokes equations, Nonlinear Analysis, 192 (2020), 111704.

[3]

J. Bourgain and N. Pavlović, Ill-posedness of the Navier-Stokes equations in a critical space in 3D, Journal of Functional Analysis, 255 (2008), 2233-2247. 

[4]

T. Buckmaster and V. Vicol, Nonuniqueness of weak solutions to the Navier-Stokes equation, Annals of Mathematics, 189 (2019), 101-144.  doi: 10.4007/annals.2019.189.1.3.

[5]

H. Cabannes, Theoretical Magnetofluiddynamics, Academic Press, New York and London, 1970.

[6]

R. E. CaflischI. Klapper and G. Steele, Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD, Communications in Mathematical Physics, 184 (1997), 443-455.  doi: 10.1007/s002200050067.

[7]

M. Cannone, Ondelettes, Paraproduits et Navier-Stokes, Paris 9, 1995.

[8]

C. Cao and E. S. Titi, Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor, Archive for Rational Mechanics and Analysis, 202 (2011), 919-932.  doi: 10.1007/s00205-011-0439-6.

[9]

C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, Journal of Differential Equations, 248 (2010), 2263-2274.  doi: 10.1016/j.jde.2009.09.020.

[10]

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

[11]

A. Cheskidov, S. Friedlander and R. Shvydkoy, On the energy equality for weak solutions of the 3D Navier-Stokes equations, Advances in Mathematical Fluid Mechanics. Springer, Berlin, Heidelberg, (2010), 171–175. doi: 10.1007/978-3-642-04068-9_10.

[12]

A. Cheskidov and X. Luo, Energy equality for the Navier-Stokes equations in weak-in-time Onsager spaces, Nonlinearity, 33 (2020), 1388-1403.  doi: 10.1088/1361-6544/ab60d3.

[13]

P. ConstantinW. E and E. S. Titi, Onsager's conjecture on the energy conservation for solutions of Euler's equation, Communications in Mathematical Physics, 165 (1994), 207-209.  doi: 10.1007/BF02099744.

[14]

G. Duvaut and J.-L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Archive for Rational Mechanics and Analysis, 46 (1972), 241-279.  doi: 10.1007/BF00250512.

[15]

L. EscauriazaG. Serë gin and V. Shverak, $L_{3, \infty}$-solutions of the Navier-Stokes equations and backward uniqueness, Russian Mathematical Surveys, 58 (2003), 211-250.  doi: 10.1070/RM2003v058n02ABEH000609.

[16]

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

[17]

C. He and Y. Wang, Limiting case for the regularity criterion of the Navier-Stokes equations and the magnetohydrodynamic equations, Science China Mathematics, 53 (2010), 1767-1774.  doi: 10.1007/s11425-010-3135-3.

[18]

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

[19]

C. He and Z. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, Journal of Functional Analysis, 227 (2005), 113-152.  doi: 10.1016/j.jfa.2005.06.009.

[20]

P. Isett, A proof of Onsager's conjecture, Annals of Mathematics, 188 (2018), 871-963.  doi: 10.4007/annals.2018.188.3.4.

[21]

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.

[22]

H. Kim and H. Kozono, Interior regularity criteria in weak spaces for the Navier-Stokes equations, Manuscripta Mathematica, 115 (2004), 85-100.  doi: 10.1007/s00229-004-0484-7.

[23]

J.-M. Kim, The energy conservations and lower bounds for possible singular solutions to the 3D incompressible MHD equations, Acta Math. Sci. Ser. B (Engl. Ed.), 40 (2020), 237-244.  doi: 10.1007/s10473-020-0116-x.

[24]

H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Advances in Mathematics, 157 (2001), 22-35. 

[25]

H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations, Math. Z., 235 (2000), 173-194.  doi: 10.1007/s002090000130.

[26]

I. Kukavica, Role of the pressure for validity of the energy equality for solutions of the Navier-Stokes equation, Journal of Dynamics and Differential Equations, 18 (2006), 461-482.  doi: 10.1007/s10884-006-9010-9.

[27]

T. M. Leslie and R. Shvydkoy, The energy measure for the Euler and Navier-Stokes equations, Archive for Rational Mechanics and Analysis, 230 (2018), 459-492.  doi: 10.1007/s00205-018-1250-4.

[28]

T. M. Leslie and R. Shvydkoy, Conditions implying energy equality for weak solutions of the Navier-Stokes equations, SIAM Journal on Mathematical Analysis, 50 (2018), 870-890.  doi: 10.1137/16M1104147.

[29]

J. L. Lions, Sur la régularité et l'unicité des solutions turbulentes des équations de Navier-Stokes, Rendiconti del Seminario Matematico della Universita di Padova, 30 (1960), 16-23. 

[30]

E. Miller, A regularity criterion for the Navier-Stokes equation involving only the middle eigenvalue of the strain tensor, Archive for Rational Mechanics and Analysis, 235 (2020), 99-139.  doi: 10.1007/s00205-019-01419-z.

[31]

L. Onsager, Statistical Hydrodynamics, Nuovo Cimento (Supplemento), 6 (1949), 279-287.  doi: 10.1007/BF02780991.

[32]

G. Prodi, Un teorema di unicita per le equazioni di Navier-Stokes, Annali di Matematica pura ed Applicata, 48 (1959), 173-182.  doi: 10.1007/BF02410664.

[33]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Communications on Pure and Applied Mathematics, 36 (1983), 635-664.  doi: 10.1002/cpa.3160360506.

[34]

J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Archive for Rational Mechanics and Analysis, 9 (1962), 187-195.  doi: 10.1007/BF00253344.

[35]

M. Shinbrot, The energy equation for the Navier-Stokes system, SIAM Journal on Mathematical Analysis, 5 (1974), 948-954.  doi: 10.1137/0505092.

[36]

W. Tan and Z. Yin, The energy measure for the Navier-Stokes and Euler equations and some applications, Submitted.

[37]

B. Wang, Ill-posedness for the Navier-Stokes equations in critical Besov spaces $\dot{B}_{\infty, q}^{-1}$, Advances in Mathematics, 268 (2015), 350-372.  doi: 10.1016/j.aim.2014.09.024.

[38]

W. Wang and Z. Zhang, On the interior regularity criteria and the number of singular points to the Navier-Stokes equations, Journal d'Analyse Mathematique, 123 (2014), 139-170.  doi: 10.1007/s11854-014-0016-7.

[39]

Y. Yong and Q. S. Jiu, Energy equality and uniqueness of weak solutions to MHD equations in $L^\infty(0, T; L^n (\Omega))$, Acta Mathematica Sinica, English Series, 25 (2009), 803-814.  doi: 10.1007/s10114-009-7214-8.

[40]

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

[41]

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.

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