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

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

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.

Citation: Feng Cheng, Chao-Jiang Xu. On the Gevrey regularity of solutions to the 3D ideal MHD equations. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6485-6506. doi: 10.3934/dcds.2019281
References:
[1]

J. T. BealT. 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.

[2]

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.

[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.

[4]

M. CannoneQ. 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.

[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.

[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.

[7]

V. K. KalantarovB. 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.

[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.

[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.

[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. 
[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.

[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.

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

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

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

J. T. BealT. 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.

[2]

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.

[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.

[4]

M. CannoneQ. 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.

[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.

[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.

[7]

V. K. KalantarovB. 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.

[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.

[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.

[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. 
[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.

[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.

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

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

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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