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

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.

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

Q. Duan, Y. Xiao and Z. Xin, On the vanishing dissipation limit for the incompressible MHD equations on bounded domains, Preprint, 2020. Google Scholar

[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.  Google Scholar

[6]

C. FeffermanD. McCormickJ. 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.  Google Scholar

[7]

O. Guès, Problème mixte hyperbolique quasi-linéaire caractéristique, Commun. Partial Differ.Equ., 15 (1990), 595-645.  doi: 10.1080/03605309908820701.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

J. LiW. 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.  Google Scholar

[11]

F. LinL. 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.  Google Scholar

[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.  Google Scholar

[13]

C.-J. LiuF. 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.  Google Scholar

[14]

C.-J. LiuF. Xie and T. Yang, Justification of Prandtl ansatz for MHD boundary layer, SIAM J. Math. Anal., 51 (2019), 2748-2791.  doi: 10.1137/18M1219618.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

X. RenJ. WuZ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

Y. WangZ. 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.  Google Scholar

[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.  Google Scholar

[23]

Y. L. XiaoZ. 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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

Q. Duan, Y. Xiao and Z. Xin, On the vanishing dissipation limit for the incompressible MHD equations on bounded domains, Preprint, 2020. Google Scholar

[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.  Google Scholar

[6]

C. FeffermanD. McCormickJ. 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.  Google Scholar

[7]

O. Guès, Problème mixte hyperbolique quasi-linéaire caractéristique, Commun. Partial Differ.Equ., 15 (1990), 595-645.  doi: 10.1080/03605309908820701.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

J. LiW. 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.  Google Scholar

[11]

F. LinL. 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.  Google Scholar

[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.  Google Scholar

[13]

C.-J. LiuF. 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.  Google Scholar

[14]

C.-J. LiuF. Xie and T. Yang, Justification of Prandtl ansatz for MHD boundary layer, SIAM J. Math. Anal., 51 (2019), 2748-2791.  doi: 10.1137/18M1219618.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

X. RenJ. WuZ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

Y. WangZ. 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.  Google Scholar

[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.  Google Scholar

[23]

Y. L. XiaoZ. 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.  Google Scholar

[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.  Google Scholar

[1]

Zhigang Wang. Vanishing viscosity limit of the rotating shallow water equations with far field vacuum. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 311-328. doi: 10.3934/dcds.2018015

[2]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero, Michael Z. Zgurovsky. Strong attractors for vanishing viscosity approximations of non-Newtonian suspension flows. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1155-1176. doi: 10.3934/dcdsb.2018146

[3]

Hua Chen, Jian-Meng Li, Kelei Wang. On the vanishing viscosity limit of a chemotaxis model. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1963-1987. doi: 10.3934/dcds.2020101

[4]

Xin Zhong. Singularity formation to the two-dimensional non-barotropic non-resistive magnetohydrodynamic equations with zero heat conduction in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1083-1096. doi: 10.3934/dcdsb.2019209

[5]

M. Bulíček, P. Kaplický. Incompressible fluids with shear rate and pressure dependent viscosity: Regularity of steady planar flows. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 41-50. doi: 10.3934/dcdss.2008.1.41

[6]

Thomas Strömberg. A system of the Hamilton--Jacobi and the continuity equations in the vanishing viscosity limit. Communications on Pure & Applied Analysis, 2011, 10 (2) : 479-506. doi: 10.3934/cpaa.2011.10.479

[7]

Pierluigi Colli, Gianni Gilardi, Pavel Krejčí, Jürgen Sprekels. A vanishing diffusion limit in a nonstandard system of phase field equations. Evolution Equations & Control Theory, 2014, 3 (2) : 257-275. doi: 10.3934/eect.2014.3.257

[8]

Quansen Jiu, Jitao Liu. Global regularity for the 3D axisymmetric MHD Equations with horizontal dissipation and vertical magnetic diffusion. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 301-322. doi: 10.3934/dcds.2015.35.301

[9]

Jishan Fan, Tohru Ozawa. Regularity criteria for the 2D MHD system with horizontal dissipation and horizontal magnetic diffusion. Kinetic & Related Models, 2014, 7 (1) : 45-56. doi: 10.3934/krm.2014.7.45

[10]

Jishan Fan, Tohru Ozawa. A regularity criterion for 3D density-dependent MHD system with zero viscosity. Conference Publications, 2015, 2015 (special) : 395-399. doi: 10.3934/proc.2015.0395

[11]

Seung-Yeal Ha, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. Uniform stability and mean-field limit for the augmented Kuramoto model. Networks & Heterogeneous Media, 2018, 13 (2) : 297-322. doi: 10.3934/nhm.2018013

[12]

Stefano Bianchini, Alberto Bressan. A case study in vanishing viscosity. Discrete & Continuous Dynamical Systems, 2001, 7 (3) : 449-476. doi: 10.3934/dcds.2001.7.449

[13]

Umberto Mosco, Maria Agostina Vivaldi. Vanishing viscosity for fractal sets. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1207-1235. doi: 10.3934/dcds.2010.28.1207

[14]

Gyungsoo Woo, Young-Sam Kwon. Incompressible limit for the full magnetohydrodynamics flows under Strong Stratification on unbounded domains. Communications on Pure & Applied Analysis, 2014, 13 (1) : 135-155. doi: 10.3934/cpaa.2014.13.135

[15]

Colette Guillopé, Zaynab Salloum, Raafat Talhouk. Regular flows of weakly compressible viscoelastic fluids and the incompressible limit. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1001-1028. doi: 10.3934/dcdsb.2010.14.1001

[16]

Wenjun Wang, Lei Yao. Vanishing viscosity limit to rarefaction waves for the full compressible fluid models of Korteweg type. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2331-2350. doi: 10.3934/cpaa.2014.13.2331

[17]

Jing Wang, Lining Tong. Vanishing viscosity limit of 1d quasilinear parabolic equation with multiple boundary layers. Communications on Pure & Applied Analysis, 2019, 18 (2) : 887-910. doi: 10.3934/cpaa.2019043

[18]

Alberto Bressan, Yilun Jiang. The vanishing viscosity limit for a system of H-J equations related to a debt management problem. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 793-824. doi: 10.3934/dcdss.2018050

[19]

Mihaï Bostan. Asymptotic behavior for the Vlasov-Poisson equations with strong uniform magnetic field and general initial conditions. Kinetic & Related Models, 2020, 13 (3) : 531-548. doi: 10.3934/krm.2020018

[20]

Wojciech M. Zajączkowski. Stability of axially-symmetric solutions to incompressible magnetohydrodynamics with no azimuthal velocity and with only azimuthal magnetic field. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1447-1482. doi: 10.3934/cpaa.2019070

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (128)
  • HTML views (171)
  • Cited by (0)

Other articles
by authors

[Back to Top]