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Subellipticity of some complex vector fields related to the Witten Laplacian
Uniform regularity and vanishing viscosity limit for the incompressible non-resistive MHD system with TMF
1. | School of Mathematical Sciences, Institute of Natural Sciences, Center of Applied Mathematics, MOE-LSC and SHL-MAC, Shanghai Jiao Tong University, Shanghai, 200240, China |
2. | Hong Kong Institute for Advanced Study, City University of Hong Kong, Hong Kong, China |
3. | School of Mathematical Sciences and LSC-MOE, Shanghai Jiao Tong University, Shanghai, 200240, China |
4. | Department of Mathematics, City University of Hong Kong, Hong Kong, China |
This paper is concerned with the vanishing viscosity limit for the incompressible MHD system without magnetic diffusion effect in the half space under the influence of a transverse magnetic field on the boundary. We prove that the solution to the incompressible MHD system is uniformly bounded in both conormal Sobolev norm and $ L^\infty $ norm in a fixed time interval independent of the viscosity coefficient. As a direct consequence, the inviscid limit from the viscous MHD system to the ideal MHD system is established in $ L^\infty $-norm. In addition, the analysis shows that the boundary layer effect is weak because of the transverse magnetic field.
References:
[1] |
H. Abidi and P. Zhang,
On the global solution of a 3-D MHD system with initial data near equilibrium, Commun. Pure Appl. Math., 70 (2017), 1509-1561.
doi: 10.1002/cpa.21645. |
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Y. Cai and Z. Lei,
Global well-Posedness of the incompressible magnetohydrodynamics, Arch. Ration. Mech. Anal., 228 (2018), 969-993.
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J. Y. Chemin, D. S. McCormick, J. C. Robinson and J. L. Rodrigo, Local existence for the non-resistive MHD equations in Besov spaces, Adv. Math., 286 (2016), 1-31.
doi: 10.1016/j. aim. 2015.09.004. |
[4] |
Q. Duan, Y. Xiao and Z. Xin, On the vanishing dissipation limit for the incompressible MHD equations on bounded domains, Preprint, 2020. |
[5] |
C. Fefferman, D. McCormick, J. Robinson and J. Rodrigo, Higher order commutator estimates and local existence for the non-resistive MHD equations and related models, J. Funct. Anal., 267(2014), 1035-1056.
doi: 10.1016/j. jfa. 2014.03.021. |
[6] |
C. Fefferman, D. McCormick, J. Robinson and J. Rodrigo,
Local existence for the non-resistive MHD equations in nearly optimal Sobolev spaces, Arch. Ration. Mech. Anal., 223 (2017), 677-691.
doi: 10.1007/s00205-016-1042-7. |
[7] |
O. Guès,
Problème mixte hyperbolique quasi-linéaire caractéristique, Commun. Partial Differ.Equ., 15 (1990), 595-645.
doi: 10.1080/03605309908820701. |
[8] |
L. B. He, L. Xu and P. Yu, On global dynamics of three dimensional magnetohydrodynamics: nonlinear stability of Alfvén waves, Ann. PDE, 5 (2018), 105pp.
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[9] |
Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Differential Equations, 259 (2015), 3202-3215.
doi: 10.1016/j. jde. 2015.04.017. |
[10] |
J. Li, W. Tan and Z. Yin,
Local existence and uniqueness for the non-resistive MHD equations in homogeneous Besov spaces, Adv. Math., 317 (2017), 786-798.
doi: 10.1016/j.aim.2017.07.013. |
[11] |
F. Lin, L. Xu and P. Zhang,
Global small solutions of 2-D incompressible MHD system, J. Differential Equations, 259 (2015), 5440-5485.
doi: 10.1016/j.jde.2015.06.034. |
[12] |
C. -J. Liu, D. Wang, F. Xie and T. Yang, Magnetic effects on the solvability of 2D MHD boundary layer equations without resistivity in Sobolev spaces, J. Funct. Anal., 279 (2020), 108637, 45pp.
doi: 10.1016/j. jfa. 2020.108637. |
[13] |
C.-J. Liu, F. Xie and T. Yang,
MHD boundary layers in Sobolev spaces without monotonicity. I. Well-posedness theory, Commun. Pure Appl. Math., 72 (2019), 63-121.
doi: 10.1002/cpa.21763. |
[14] |
C.-J. Liu, F. Xie and T. Yang,
Justification of Prandtl ansatz for MHD boundary layer, SIAM J. Math. Anal., 51 (2019), 2748-2791.
doi: 10.1137/18M1219618. |
[15] |
N. Masmoudi and F. Rousset,
Uniform regularity for the Navier-Stokes equation with Navier boundary condition, Arch. Ration. Mech. Anal., 203 (2012), 529-575.
doi: 10.1007/s00205-011-0456-5. |
[16] |
N. Masmoudi and F. Rousset,
Uniform regularity and vanishing viscosity limit for the free surface Navier-Stokes equations, Arch. Ration. Mech. Anal., 223 (2017), 301-417.
doi: 10.1007/s00205-016-1036-5. |
[17] |
M. Paddick,
The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions, Discret. Contin. Dyn. Syst., 36 (2016), 2673-2709.
doi: 10.3934/dcds.2016.36.2673. |
[18] |
X. Ren, J. Wu, Z. Xiang and Z. Zhang,
Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion, J. Funct. Anal., 267 (2014), 503-541.
doi: 10.1016/j.jfa.2014.04.020. |
[19] |
R. Wan,
On the uniqueness for the 2D MHD equations without magnetic diffusion, Nonlinear Anal. Real World Appl., 30 (2016), 32-40.
doi: 10.1016/j.nonrwa.2015.11.006. |
[20] |
Y. Wang,
Uniform regularity and vanishing dissipation limit for the full compressible Navier-Stokes system in three dimensional bounded domain, Arch. Ration. Mech. Anal., 221 (2015), 4123-4191.
doi: 10.1007/s00205-016-0989-8. |
[21] |
Y. Wang, Z. P. Xin and Y. Yong,
Uniform regularity and vanishing viscosity limit for the compressible Navier-Stokes with general Navier-slip boundary conditions in 3-dimensional domains, SIAM J. Math. Anal., 47 (2015), 4123-4191.
doi: 10.1137/151003520. |
[22] |
D. Wei and Z. Zhang,
Global well-posedness of the MHD equations in a homogeneous magnetic field, Anal. PDE, 10 (2017), 1361-1406.
doi: 10.2140/apde.2017.10.1361. |
[23] |
Y. L. Xiao, Z. P. Xin and J. H. Wu,
Vanishing viscosity limit for the 3D magnetohydrodynamic system with a slip boundary condition, J. Funct. Anal., 257 (2009), 3375-3394.
doi: 10.1016/j.jfa.2009.09.010. |
[24] |
L. Xu and P. Zhang,
Global small solutions to three-dimensional incompressible magnetohydrodynamical system, SIAM J. Math. Anal., 47 (2015), 26-65.
doi: 10.1137/14095515X. |
show all references
References:
[1] |
H. Abidi and P. Zhang,
On the global solution of a 3-D MHD system with initial data near equilibrium, Commun. Pure Appl. Math., 70 (2017), 1509-1561.
doi: 10.1002/cpa.21645. |
[2] |
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. |
[3] |
J. Y. Chemin, D. S. McCormick, J. C. Robinson and J. L. Rodrigo, Local existence for the non-resistive MHD equations in Besov spaces, Adv. Math., 286 (2016), 1-31.
doi: 10.1016/j. aim. 2015.09.004. |
[4] |
Q. Duan, Y. Xiao and Z. Xin, On the vanishing dissipation limit for the incompressible MHD equations on bounded domains, Preprint, 2020. |
[5] |
C. Fefferman, D. McCormick, J. Robinson and J. Rodrigo, Higher order commutator estimates and local existence for the non-resistive MHD equations and related models, J. Funct. Anal., 267(2014), 1035-1056.
doi: 10.1016/j. jfa. 2014.03.021. |
[6] |
C. Fefferman, D. McCormick, J. Robinson and J. Rodrigo,
Local existence for the non-resistive MHD equations in nearly optimal Sobolev spaces, Arch. Ration. Mech. Anal., 223 (2017), 677-691.
doi: 10.1007/s00205-016-1042-7. |
[7] |
O. Guès,
Problème mixte hyperbolique quasi-linéaire caractéristique, Commun. Partial Differ.Equ., 15 (1990), 595-645.
doi: 10.1080/03605309908820701. |
[8] |
L. B. He, L. Xu and P. Yu, On global dynamics of three dimensional magnetohydrodynamics: nonlinear stability of Alfvén waves, Ann. PDE, 5 (2018), 105pp.
doi: 10.1007/s40818-017-0041-9. |
[9] |
Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Differential Equations, 259 (2015), 3202-3215.
doi: 10.1016/j. jde. 2015.04.017. |
[10] |
J. Li, W. Tan and Z. Yin,
Local existence and uniqueness for the non-resistive MHD equations in homogeneous Besov spaces, Adv. Math., 317 (2017), 786-798.
doi: 10.1016/j.aim.2017.07.013. |
[11] |
F. Lin, L. Xu and P. Zhang,
Global small solutions of 2-D incompressible MHD system, J. Differential Equations, 259 (2015), 5440-5485.
doi: 10.1016/j.jde.2015.06.034. |
[12] |
C. -J. Liu, D. Wang, F. Xie and T. Yang, Magnetic effects on the solvability of 2D MHD boundary layer equations without resistivity in Sobolev spaces, J. Funct. Anal., 279 (2020), 108637, 45pp.
doi: 10.1016/j. jfa. 2020.108637. |
[13] |
C.-J. Liu, F. Xie and T. Yang,
MHD boundary layers in Sobolev spaces without monotonicity. I. Well-posedness theory, Commun. Pure Appl. Math., 72 (2019), 63-121.
doi: 10.1002/cpa.21763. |
[14] |
C.-J. Liu, F. Xie and T. Yang,
Justification of Prandtl ansatz for MHD boundary layer, SIAM J. Math. Anal., 51 (2019), 2748-2791.
doi: 10.1137/18M1219618. |
[15] |
N. Masmoudi and F. Rousset,
Uniform regularity for the Navier-Stokes equation with Navier boundary condition, Arch. Ration. Mech. Anal., 203 (2012), 529-575.
doi: 10.1007/s00205-011-0456-5. |
[16] |
N. Masmoudi and F. Rousset,
Uniform regularity and vanishing viscosity limit for the free surface Navier-Stokes equations, Arch. Ration. Mech. Anal., 223 (2017), 301-417.
doi: 10.1007/s00205-016-1036-5. |
[17] |
M. Paddick,
The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions, Discret. Contin. Dyn. Syst., 36 (2016), 2673-2709.
doi: 10.3934/dcds.2016.36.2673. |
[18] |
X. Ren, J. Wu, Z. Xiang and Z. Zhang,
Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion, J. Funct. Anal., 267 (2014), 503-541.
doi: 10.1016/j.jfa.2014.04.020. |
[19] |
R. Wan,
On the uniqueness for the 2D MHD equations without magnetic diffusion, Nonlinear Anal. Real World Appl., 30 (2016), 32-40.
doi: 10.1016/j.nonrwa.2015.11.006. |
[20] |
Y. Wang,
Uniform regularity and vanishing dissipation limit for the full compressible Navier-Stokes system in three dimensional bounded domain, Arch. Ration. Mech. Anal., 221 (2015), 4123-4191.
doi: 10.1007/s00205-016-0989-8. |
[21] |
Y. Wang, Z. P. Xin and Y. Yong,
Uniform regularity and vanishing viscosity limit for the compressible Navier-Stokes with general Navier-slip boundary conditions in 3-dimensional domains, SIAM J. Math. Anal., 47 (2015), 4123-4191.
doi: 10.1137/151003520. |
[22] |
D. Wei and Z. Zhang,
Global well-posedness of the MHD equations in a homogeneous magnetic field, Anal. PDE, 10 (2017), 1361-1406.
doi: 10.2140/apde.2017.10.1361. |
[23] |
Y. L. Xiao, Z. P. Xin and J. H. Wu,
Vanishing viscosity limit for the 3D magnetohydrodynamic system with a slip boundary condition, J. Funct. Anal., 257 (2009), 3375-3394.
doi: 10.1016/j.jfa.2009.09.010. |
[24] |
L. Xu and P. Zhang,
Global small solutions to three-dimensional incompressible magnetohydrodynamical system, SIAM J. Math. Anal., 47 (2015), 26-65.
doi: 10.1137/14095515X. |
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