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November  2019, 39(11): 6485-6506. doi: 10.3934/dcds.2019281

On the Gevrey regularity of solutions to the 3D ideal MHD equations

Feng Cheng 1,, and  Chao-Jiang Xu 2,3,

1. 

Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, 430062 Wuhan, China
2. 

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, 211106 Nanjing, China
3. 

Université de Rouen, CNRS UMR 6085, Laboratoire de Mathématiques, 76801 Saint-Etienne du Rouvray, France

* Corresponding author: Feng Cheng

Received  December 2018 Revised  May 2019 Published  August 2019

In this paper, we prove the propagation of the Gevrey regularity of solutions to the three-dimensional incompressible ideal magnetohydrodynamics (MHD) equations. We also obtain an uniform estimate of Gevrey radius for the solution of MHD equation.

Keywords: Gevrey class regularity, incompressible magnetohydrodynamic equation, analyticity.
Mathematics Subject Classification: Primary: P35Q35; Secondary: 76B03, 76W05.
Citation: Feng Cheng, Chao-Jiang Xu. On the Gevrey regularity of solutions to the 3D ideal MHD equations. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6485-6506. doi: 10.3934/dcds.2019281
References:
[1]

J. T. Beal, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Communications in Mathematical Physics, 94 (1984), 61-66.  doi: 10.1007/BF01212349.  Google Scholar

[2]

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

[3]

Y. Cai and Z. Lei, Global well-posedness of the incompressible magnetohydrodynamics, Arch. Ration. Mech. Anal., 228 (2018), 969-993.  doi: 10.1007/s00205-017-1210-4.  Google Scholar

[4]

M. Cannone, Q. L. Chen and C. X. Miao, A losing estimate for the ideal MHD equations with application to blow-up criterion, SIAM Journal on Mathematical Analysis, 38 (2007), 1847-1859.  doi: 10.1137/060652002.  Google Scholar

[5]

C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal, 87 (1989), 359-369.  doi: 10.1016/0022-1236(89)90015-3.  Google Scholar

[6]

L.-B. He, L. Xu and P. Yu, On global dynamics of three dimensional magnetohydrodynamics: Nonlinear Stability of Alfvén waves, Annals of PDE, 4 (2018), Art. 5,105 pp. doi: 10.1007/s40818-017-0041-9.  Google Scholar

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V. K. Kalantarov, B. Levant and E. S. Titi, Gevrey regularity for the global attractor of the 3D Navier-Stokes-Voight equations, J. Nonlinear Sci., 19 (2009), 133-152.  doi: 10.1007/s00332-008-9029-7.  Google Scholar

[8]

T. Kato and C. Y. Lai, Nonlinear evolution equations and the Euler flow, J. Funct. Anal., 56 (1984), 15-28.  doi: 10.1016/0022-1236(84)90024-7.  Google Scholar

[9]

I. Kukavica and V. Vicol, On the radius of analyticity of solutions to the three-dimensional Euler equations, Proc. Amer. Math. Soc, 137 (2009), 669-677.  doi: 10.1090/S0002-9939-08-09693-7.  Google Scholar

[10] L. D. Laudau and E. M. Lifshitz, Electrondynamics of Continuous Media, Course of Theoretical Physics, Vol. 8. Pergamon Press, Oxford-London-New York-Paris, Addison-Wesley Publishing Co., Inc., Reading, Mass, 1960.   Google Scholar
[11]

C. D. Levermore and M. Oliver, Analyticity of solutions for a generalized Euler equation, J. Differential Equations, 133 (1997), 321-339.  doi: 10.1006/jdeq.1996.3200.  Google Scholar

[12]

F. C. Li and Z. P. Zhang, Zero viscosity-resistivity limit for the 3D incompressible magnetohydrodynamic equations in Gevrey class, Discrete Contin. Dyn. Syst., 38 (2018), 4279-4304.  doi: 10.3934/dcds.2018187.  Google Scholar

[13] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, 2002.   Google Scholar
[14]

S. Kim, Gevrey class regularity of the magnetohydrodynamics equations, ANZIAM J., 43 (2002), 397-408.  doi: 10.1017/S1446181100012591.  Google Scholar

[15]

P. Secchi, On the equations of ideal incompressible magneto-hydrodynamics, Rendiconti del Seminario Matematico della Universite di Padova, 90 (1993), 103-119.   Google Scholar

[16]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664.  doi: 10.1002/cpa.3160360506.  Google Scholar

[17]

R. Temam, On the Euler equations of incompressible perfect fluids, J. Funct. Anal., 20 (1975), 32-43.  doi: 10.1016/0022-1236(75)90052-X.  Google Scholar

[18]

Y.-Z. Wang and P. F. Li, Global existence of three dimensional incompressible MHD flows, Math. Methods Appl. Sci., 39 (2016), 4246-4256.  doi: 10.1002/mma.3862.  Google Scholar

[19]

S. K. Weng, On analyticity and temporal decay rates of solutions to the viscous resistive Hall-MHD system, J. Differential Equations, 260 (2016), 6504-6524.  doi: 10.1016/j.jde.2016.01.003.  Google Scholar

[20]

J. Wu, Bounds and new approaches for the 3D MHD equations, J. Nonlinear Sci., 12 (2002), 395-413.  doi: 10.1007/s00332-002-0486-0.  Google Scholar

[21]

Y. J. Yu and K. T. Li, Existence of solutions for the MHD-Leray-alpha equations and their relations to the MHD equations, J. Math. Anal. Appl., 329 (2007), 298-326.  doi: 10.1016/j.jmaa.2006.06.039.  Google Scholar

[22]

Z. F. Zhang and X. F. Liu, On the blow-up criterion of smooth solutions to the 3D ideal MHD equations, Acta Math. Appl. Sin. Engl. Ser., 20 (2004), 695-700.  doi: 10.1007/s10255-004-0207-6.  Google Scholar

[23]

C. D. Zhao and B. Li, Analyticity of the global attractor for the 3D regularized MHD equations, Electron. J. Diff. Equ., 2016 (2016), 1-20.  Google Scholar

show all references

References:
[1]

J. T. Beal, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Communications in Mathematical Physics, 94 (1984), 61-66.  doi: 10.1007/BF01212349.  Google Scholar

[2]

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

[3]

Y. Cai and Z. Lei, Global well-posedness of the incompressible magnetohydrodynamics, Arch. Ration. Mech. Anal., 228 (2018), 969-993.  doi: 10.1007/s00205-017-1210-4.  Google Scholar

[4]

M. Cannone, Q. L. Chen and C. X. Miao, A losing estimate for the ideal MHD equations with application to blow-up criterion, SIAM Journal on Mathematical Analysis, 38 (2007), 1847-1859.  doi: 10.1137/060652002.  Google Scholar

[5]

C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal, 87 (1989), 359-369.  doi: 10.1016/0022-1236(89)90015-3.  Google Scholar

[6]

L.-B. He, L. Xu and P. Yu, On global dynamics of three dimensional magnetohydrodynamics: Nonlinear Stability of Alfvén waves, Annals of PDE, 4 (2018), Art. 5,105 pp. doi: 10.1007/s40818-017-0041-9.  Google Scholar

[7]

V. K. Kalantarov, B. Levant and E. S. Titi, Gevrey regularity for the global attractor of the 3D Navier-Stokes-Voight equations, J. Nonlinear Sci., 19 (2009), 133-152.  doi: 10.1007/s00332-008-9029-7.  Google Scholar

[8]

T. Kato and C. Y. Lai, Nonlinear evolution equations and the Euler flow, J. Funct. Anal., 56 (1984), 15-28.  doi: 10.1016/0022-1236(84)90024-7.  Google Scholar

[9]

I. Kukavica and V. Vicol, On the radius of analyticity of solutions to the three-dimensional Euler equations, Proc. Amer. Math. Soc, 137 (2009), 669-677.  doi: 10.1090/S0002-9939-08-09693-7.  Google Scholar

[10] L. D. Laudau and E. M. Lifshitz, Electrondynamics of Continuous Media, Course of Theoretical Physics, Vol. 8. Pergamon Press, Oxford-London-New York-Paris, Addison-Wesley Publishing Co., Inc., Reading, Mass, 1960.   Google Scholar
[11]

C. D. Levermore and M. Oliver, Analyticity of solutions for a generalized Euler equation, J. Differential Equations, 133 (1997), 321-339.  doi: 10.1006/jdeq.1996.3200.  Google Scholar

[12]

F. C. Li and Z. P. Zhang, Zero viscosity-resistivity limit for the 3D incompressible magnetohydrodynamic equations in Gevrey class, Discrete Contin. Dyn. Syst., 38 (2018), 4279-4304.  doi: 10.3934/dcds.2018187.  Google Scholar

[13] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, 2002.   Google Scholar
[14]

S. Kim, Gevrey class regularity of the magnetohydrodynamics equations, ANZIAM J., 43 (2002), 397-408.  doi: 10.1017/S1446181100012591.  Google Scholar

[15]

P. Secchi, On the equations of ideal incompressible magneto-hydrodynamics, Rendiconti del Seminario Matematico della Universite di Padova, 90 (1993), 103-119.   Google Scholar

[16]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664.  doi: 10.1002/cpa.3160360506.  Google Scholar

[17]

R. Temam, On the Euler equations of incompressible perfect fluids, J. Funct. Anal., 20 (1975), 32-43.  doi: 10.1016/0022-1236(75)90052-X.  Google Scholar

[18]

Y.-Z. Wang and P. F. Li, Global existence of three dimensional incompressible MHD flows, Math. Methods Appl. Sci., 39 (2016), 4246-4256.  doi: 10.1002/mma.3862.  Google Scholar

[19]

S. K. Weng, On analyticity and temporal decay rates of solutions to the viscous resistive Hall-MHD system, J. Differential Equations, 260 (2016), 6504-6524.  doi: 10.1016/j.jde.2016.01.003.  Google Scholar

[20]

J. Wu, Bounds and new approaches for the 3D MHD equations, J. Nonlinear Sci., 12 (2002), 395-413.  doi: 10.1007/s00332-002-0486-0.  Google Scholar

[21]

Y. J. Yu and K. T. Li, Existence of solutions for the MHD-Leray-alpha equations and their relations to the MHD equations, J. Math. Anal. Appl., 329 (2007), 298-326.  doi: 10.1016/j.jmaa.2006.06.039.  Google Scholar

[22]

Z. F. Zhang and X. F. Liu, On the blow-up criterion of smooth solutions to the 3D ideal MHD equations, Acta Math. Appl. Sin. Engl. Ser., 20 (2004), 695-700.  doi: 10.1007/s10255-004-0207-6.  Google Scholar

[23]

C. D. Zhao and B. Li, Analyticity of the global attractor for the 3D regularized MHD equations, Electron. J. Diff. Equ., 2016 (2016), 1-20.  Google Scholar

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