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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 |
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.
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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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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