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July & August  2021, 20(7&8): 2725-2750. doi: 10.3934/cpaa.2021073

Uniform regularity and vanishing viscosity limit for the incompressible non-resistive MHD system with TMF

Cheng-Jie Liu 1,2, , Feng Xie 3, and  Tong Yang 4,,

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

* Corresponding author

Dedicated to Professor Shuxing Chen on the occasion of his 80th birthday

Received  January 2021 Revised  March 2021 Published  July & August 2021 Early access  April 2021

Fund Project: The research of C.-J. Liu was supported by National Natural Science Foundation of China (Grant No. 11743009, 11801364), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDA25010402), Shanghai Sailing Program (Grant No. 18YF1411700), a startup grant from Shanghai Jiao Tong University (Grant No. WF220441906), and the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning. F. Xie was supported by National Natural Science Foundation of China No.11831003 and Shanghai Science and Technology Innovation Action Plan No. 20JC1413000. The research of T. Yang was supported by the General Research Fund of Hong Kong Project No 11304419. The first author's research was also supported by Hong Kong Institute for Advanced Study, No. 9360157

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.

Keywords: Incompressible non-resistive MHD flows, uniform regularity, vanishing viscosity limit, transverse magnetic field.
Mathematics Subject Classification: Primary: 35G33, 35M31, 76N20.
Citation: Cheng-Jie Liu, Feng Xie, Tong Yang. Uniform regularity and vanishing viscosity limit for the incompressible non-resistive MHD system with TMF. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2725-2750. doi: 10.3934/cpaa.2021073
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.

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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