September 2018, 38(9): 4279-4304. doi: 10.3934/dcds.2018187

Zero viscosity-resistivity limit for the 3D incompressible magnetohydrodynamic equations in Gevrey class

Department of Mathematics, Nanjing University, Nanjing 210093, China

* Corresponding author: Zhipeng Zhang

Received  May 2017 Revised  October 2017 Published  June 2018

We study the zero viscosity-resistivity limit for the 3D incompressible magnetohydrodynamic (MHD) equations in a periodic domain in the framework of Gevrey class. We first prove that there exists an interval of time, independent of the viscosity coefficient and the resistivity coefficient, for the solutions to the viscous incompressible MHD equations. Then, based on these uniform estimates, we show that the solutions of the viscous incompressible MHD equations converge to that of the ideal incompressible MHD equations as the viscosity and resistivity coefficients go to zero. Moreover, the convergence rate is also given.

Citation: Fucai Li, Zhipeng Zhang. Zero viscosity-resistivity limit for the 3D incompressible magnetohydrodynamic equations in Gevrey class. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4279-4304. doi: 10.3934/dcds.2018187
References:
[1]

R. AlexandreY.-G. WangC.-J. Xu and T. Yang, Well-posedness of the Prandtl equation in Sobolev spaces, J. Amer. Math. Soc., 28 (2015), 745-784. doi: 10.1090/S0894-0347-2014-00813-4.

[2]

A.-B. Ferrari and E.-S. Titi, Gevrey regularity for nonlinear analytic parabolic equations, Comm. Partial Differential Equations, 23 (1998), 1-16. doi: 10.1080/03605309808821336.

[3]

H. Beirão da Veiga, Vorticity and regularity for flows under the Navier boundary condition, Commun. Pure Appl. Anal., 5 (2006), 907-918. doi: 10.3934/cpaa.2006.5.907.

[4]

H. Beirão da Veiga and F. Crispo, Concerning the $W^{k,p}$ -inviscid limit for 3-D flows under a slip boundary condition, J. Math. Fluid Mech., 13 (2011), 117-135. doi: 10.1007/s00021-009-0012-3.

[5]

H. Beirão da Veiga and F. Crispo, Sharp inviscid limit results under Navier type boundary conditions. An $L^p$ theory, J. Math. Fluid Mech., 12 (2010), 397-411. doi: 10.1007/s00021-009-0295-4.

[6]

D. Biskamp, Nonlinear Magnetohydrodynamics, Cambridge University Press, Cambridge, UK, 1993. doi: 10.1017/CBO9780511599965.

[7]

F. ChengW.-X. Li and C.-J. Xu, Vanishing viscosity limit of Navier-Stokes equations in Gevrey class, Math. Methods Appl. Sci., 40 (2017), 5161-5176. doi: 10.1002/mma.4378.

[8]

P. Constantin, Note on loss of regularity for solutions of the 3-D incompressible Euler and related equations, Comm. Math. Phys., 104 (1986), 311-326. doi: 10.1007/BF01211598.

[9]

P. Constantin and C. Foias, Navier Stokes Equation, Univ. of Chicago press IL, 1988.

[10]

P. ConstantinI. Kukavica and V. Vicol, On the inviscid limit of the Navier-Stokes equations, Proc. Amer. Math. Soc., 143 (2015), 3075-3090. doi: 10.1090/S0002-9939-2015-12638-X.

[11]

G. Duvaut and J.-L. Lions, Inéquation en thermoélasticite et magnétohydrodynamique, Arch. Ration. Mech. Anal., 46 (1972), 241-279. doi: 10.1007/BF00250512.

[12]

W. E and B. Engquist, Blowup of solutions of the unsteady Prandtl's equation, Comm. Pure Appl. Math., 50 (1997), 1287-1293. doi: 10.1002/(SICI)1097-0312(199712)50:12<1287::AID-CPA4>3.0.CO;2-4.

[13]

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.

[14]

B. Franck and F. Pierre, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, 183, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.

[15]

J.-P. Freidberg, Ideal Magnetohydrodynamics, New York, London, Plenum Press, 1987.

[16]

D. Gerard-Varet, Y. Maekawa and N. Masmoudi, Gevrey stability of Prandtl expansions for 2D Navier-Stokes, arXiv: 1607.06434.

[17]

D. Gerard-Varet and E. Dormy, On the ill-posedness of the Prandtl equation, J. Amer. Math. Soc., 23 (2010), 591-609. doi: 10.1090/S0894-0347-09-00652-3.

[18]

J.-F. Gerbeau, C.-L. Bris and T. Lelièvre, Mathematical Methods for the Magnetohydrodynamics of Liquid Metals, Oxford University Press, Oxford, 2006. doi: 10.1093/acprof:oso/9780198566656.001.0001.

[19]

M. Gevrey, Sur la nature analytique des solutions des équations aux dérivées partielles, Premier Mémoire, (French) Ann. Sci. École Norm. Sup., 35 (1918), 129-190. doi: 10.24033/asens.706.

[20]

G.-M. Gie and J.-P. Kelliher, Boundary layer analysis of the Navier-Stokes equations with generalized Navier boundary conditions, J. Differential Equations, 253 (2012), 1862-1892. doi: 10.1016/j.jde.2012.06.008.

[21]

Y. Guo and T. Nguyen, A note on the Prandtl boundary layers, Comm. Pure Appl. Math., 64 (2011), 1416-1438. doi: 10.1002/cpa.20377.

[22]

T. Kato, Nonstationary flows of viscous and ideal fluids in $R^3$, J. Funct. Anal., 9 (1972), 296-305. doi: 10.1016/0022-1236(72)90003-1.

[23]

T. Kato, Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary, Seminar on Nonlinear Partial Differential Equations, (Berkeley, Calif., 1983), 85–98, Math. Sci. Res. Inst. Publ., 2, Springer, New York, 1984. doi: 10.1007/978-1-4612-1110-5_6.

[24]

J.-P. Kelliher, On Kato's conditions for vanishing viscosity, Indiana Univ. Math. J., 56 (2007), 1711-1721. doi: 10.1512/iumj.2007.56.3080.

[25]

J.-P. Kelliher, Vanishing viscosity and the accumulation of vorticity on the boundary, Commun. Math. Sci., 6 (2008), 869-880. doi: 10.4310/CMS.2008.v6.n4.a4.

[26]

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.

[27]

A. Larios and E.-S. Titi, On the higher-order global regularity of the inviscid Voigt-regularization of three-dimensional hydrodynamic models, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 603-627. doi: 10.3934/dcdsb.2010.14.603.

[28]

F.-C. Li and Z.-P. Zhang, Zero kinematic viscosity-magnetic diffusion limit of the incompressible viscous magnetohydrodynamic equations with Navier boundary conditions, arXiv: 1606.05038.

[29]

W.-X. LiD. Wu and C.-J. Xu, Gevrey class smoothing effect for the Prandtl equation, SIAM J. Math. Anal., 48 (2016), 1672-1726. doi: 10.1137/15M1020368.

[30]

C.-J. Liu and T. Yang, Ill-posedness of the Prandtl equations in Sobolev spaces around a shear flow with general decay, J. Math. Pure Appl., 108 (2017), 150-162. doi: 10.1016/j.matpur.2016.10.014.

[31]

C.-J. Liu, F. Xie and T. Yang, MHD boundary layers theory in Sobolev spaces without monotonicity. Ⅰ. Well-posedness theory, arXiv: 1611.05815v4.

[32]

C.-J. Liu, F. Xie and T. Yang, MHD boundary layers theory in Sobolev spaces without monotonicity. Ⅱ. Convergence theory, arXiv: 1704.00523v1.

[33]

Y. Maekawa, On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half-plane, Comm. Pure Appl. Math., 67 (2014), 1045-1128. doi: 10.1002/cpa.21516.

[34]

N. Masmoudi, Remarks about the inviscid limit of the Navier-Stokes system, Comm. Math. Phys., 270 (2007), 777-788. doi: 10.1007/s00220-006-0171-5.

[35]

N. Masmoudi and T.-K. Wong, Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods, Comm. Pure Appl. Math., 68 (2015), 1683-1741. doi: 10.1002/cpa.21595.

[36]

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.

[37]

O.-A. Oleinik and V.-N. Samokhin, Mathematical Models in Boundary Layer Theory, Chapman and Hall/CRC, 1999.

[38]

L. Prandtl, Über flüssigkeits-bewegung bei sehr kleiner reibung, Verhandlungen des III, Internationlen Mathematiker Kongresses, Heidelberg. Teubner, Leipzig, (1904), 484-491.

[39]

L. Rodino, Linear Partial Differential Operators in Gevrey Spaces, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. doi: 10.1142/9789814360036_0002.

[40]

M. Sammartino and R.-E. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. Ⅰ. Existence for Euler and Prandtl equations, Comm. Math. Phys., 192 (1998), 433-461. doi: 10.1007/s002200050304.

[41]

M. Sammartino and R.-E. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. Ⅱ. Construction of the Navier-Stokes solution, Comm. Math. Phys., 192 (1998), 463-491. doi: 10.1007/s002200050305.

[42]

R. Temam and X.-M. Wang, On the behavior of the solutions of the Navier-Stokes equations at vanishing viscosity, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 807-828.

[43]

X.-M. Wang, A Kato type theorem on zero viscosity limit of Navier-Stokes flows, Indiana Univ. Math. J., 50 (2001), 223-241. doi: 10.1512/iumj.2001.50.2098.

[44]

C. WangY.-X. Wang and Z.-F. Zhang, Zero-viscosity limit of the Navier-Stokes equations in the analytic setting, Arch. Ration. Mech. Anal., 224 (2017), 555-595. doi: 10.1007/s00205-017-1083-6.

[45]

S. Wang and Z.-P. Xin, Boundary layer problems in the viscosity-diffusion vanishing limits for the incompressible MHD systems (in Chinese), Sci. China. Math., 47 (2017), 1303-1326. doi: 10.1360/N012016-00211.

[46]

Y.-L. Xiao and Z.-P. Xin, On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition, Comm. Pure Appl. Math., 60 (2007), 1027-1055. doi: 10.1002/cpa.20187.

[47]

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.

[48]

Z.-P. Xin and L.-Q. Zhang, On the global existence of solutions to the Prandtl system, Adv. Math., 181 (2004), 88-133. doi: 10.1016/S0001-8708(03)00046-X.

show all references

References:
[1]

R. AlexandreY.-G. WangC.-J. Xu and T. Yang, Well-posedness of the Prandtl equation in Sobolev spaces, J. Amer. Math. Soc., 28 (2015), 745-784. doi: 10.1090/S0894-0347-2014-00813-4.

[2]

A.-B. Ferrari and E.-S. Titi, Gevrey regularity for nonlinear analytic parabolic equations, Comm. Partial Differential Equations, 23 (1998), 1-16. doi: 10.1080/03605309808821336.

[3]

H. Beirão da Veiga, Vorticity and regularity for flows under the Navier boundary condition, Commun. Pure Appl. Anal., 5 (2006), 907-918. doi: 10.3934/cpaa.2006.5.907.

[4]

H. Beirão da Veiga and F. Crispo, Concerning the $W^{k,p}$ -inviscid limit for 3-D flows under a slip boundary condition, J. Math. Fluid Mech., 13 (2011), 117-135. doi: 10.1007/s00021-009-0012-3.

[5]

H. Beirão da Veiga and F. Crispo, Sharp inviscid limit results under Navier type boundary conditions. An $L^p$ theory, J. Math. Fluid Mech., 12 (2010), 397-411. doi: 10.1007/s00021-009-0295-4.

[6]

D. Biskamp, Nonlinear Magnetohydrodynamics, Cambridge University Press, Cambridge, UK, 1993. doi: 10.1017/CBO9780511599965.

[7]

F. ChengW.-X. Li and C.-J. Xu, Vanishing viscosity limit of Navier-Stokes equations in Gevrey class, Math. Methods Appl. Sci., 40 (2017), 5161-5176. doi: 10.1002/mma.4378.

[8]

P. Constantin, Note on loss of regularity for solutions of the 3-D incompressible Euler and related equations, Comm. Math. Phys., 104 (1986), 311-326. doi: 10.1007/BF01211598.

[9]

P. Constantin and C. Foias, Navier Stokes Equation, Univ. of Chicago press IL, 1988.

[10]

P. ConstantinI. Kukavica and V. Vicol, On the inviscid limit of the Navier-Stokes equations, Proc. Amer. Math. Soc., 143 (2015), 3075-3090. doi: 10.1090/S0002-9939-2015-12638-X.

[11]

G. Duvaut and J.-L. Lions, Inéquation en thermoélasticite et magnétohydrodynamique, Arch. Ration. Mech. Anal., 46 (1972), 241-279. doi: 10.1007/BF00250512.

[12]

W. E and B. Engquist, Blowup of solutions of the unsteady Prandtl's equation, Comm. Pure Appl. Math., 50 (1997), 1287-1293. doi: 10.1002/(SICI)1097-0312(199712)50:12<1287::AID-CPA4>3.0.CO;2-4.

[13]

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.

[14]

B. Franck and F. Pierre, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, 183, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.

[15]

J.-P. Freidberg, Ideal Magnetohydrodynamics, New York, London, Plenum Press, 1987.

[16]

D. Gerard-Varet, Y. Maekawa and N. Masmoudi, Gevrey stability of Prandtl expansions for 2D Navier-Stokes, arXiv: 1607.06434.

[17]

D. Gerard-Varet and E. Dormy, On the ill-posedness of the Prandtl equation, J. Amer. Math. Soc., 23 (2010), 591-609. doi: 10.1090/S0894-0347-09-00652-3.

[18]

J.-F. Gerbeau, C.-L. Bris and T. Lelièvre, Mathematical Methods for the Magnetohydrodynamics of Liquid Metals, Oxford University Press, Oxford, 2006. doi: 10.1093/acprof:oso/9780198566656.001.0001.

[19]

M. Gevrey, Sur la nature analytique des solutions des équations aux dérivées partielles, Premier Mémoire, (French) Ann. Sci. École Norm. Sup., 35 (1918), 129-190. doi: 10.24033/asens.706.

[20]

G.-M. Gie and J.-P. Kelliher, Boundary layer analysis of the Navier-Stokes equations with generalized Navier boundary conditions, J. Differential Equations, 253 (2012), 1862-1892. doi: 10.1016/j.jde.2012.06.008.

[21]

Y. Guo and T. Nguyen, A note on the Prandtl boundary layers, Comm. Pure Appl. Math., 64 (2011), 1416-1438. doi: 10.1002/cpa.20377.

[22]

T. Kato, Nonstationary flows of viscous and ideal fluids in $R^3$, J. Funct. Anal., 9 (1972), 296-305. doi: 10.1016/0022-1236(72)90003-1.

[23]

T. Kato, Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary, Seminar on Nonlinear Partial Differential Equations, (Berkeley, Calif., 1983), 85–98, Math. Sci. Res. Inst. Publ., 2, Springer, New York, 1984. doi: 10.1007/978-1-4612-1110-5_6.

[24]

J.-P. Kelliher, On Kato's conditions for vanishing viscosity, Indiana Univ. Math. J., 56 (2007), 1711-1721. doi: 10.1512/iumj.2007.56.3080.

[25]

J.-P. Kelliher, Vanishing viscosity and the accumulation of vorticity on the boundary, Commun. Math. Sci., 6 (2008), 869-880. doi: 10.4310/CMS.2008.v6.n4.a4.

[26]

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.

[27]

A. Larios and E.-S. Titi, On the higher-order global regularity of the inviscid Voigt-regularization of three-dimensional hydrodynamic models, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 603-627. doi: 10.3934/dcdsb.2010.14.603.

[28]

F.-C. Li and Z.-P. Zhang, Zero kinematic viscosity-magnetic diffusion limit of the incompressible viscous magnetohydrodynamic equations with Navier boundary conditions, arXiv: 1606.05038.

[29]

W.-X. LiD. Wu and C.-J. Xu, Gevrey class smoothing effect for the Prandtl equation, SIAM J. Math. Anal., 48 (2016), 1672-1726. doi: 10.1137/15M1020368.

[30]

C.-J. Liu and T. Yang, Ill-posedness of the Prandtl equations in Sobolev spaces around a shear flow with general decay, J. Math. Pure Appl., 108 (2017), 150-162. doi: 10.1016/j.matpur.2016.10.014.

[31]

C.-J. Liu, F. Xie and T. Yang, MHD boundary layers theory in Sobolev spaces without monotonicity. Ⅰ. Well-posedness theory, arXiv: 1611.05815v4.

[32]

C.-J. Liu, F. Xie and T. Yang, MHD boundary layers theory in Sobolev spaces without monotonicity. Ⅱ. Convergence theory, arXiv: 1704.00523v1.

[33]

Y. Maekawa, On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half-plane, Comm. Pure Appl. Math., 67 (2014), 1045-1128. doi: 10.1002/cpa.21516.

[34]

N. Masmoudi, Remarks about the inviscid limit of the Navier-Stokes system, Comm. Math. Phys., 270 (2007), 777-788. doi: 10.1007/s00220-006-0171-5.

[35]

N. Masmoudi and T.-K. Wong, Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods, Comm. Pure Appl. Math., 68 (2015), 1683-1741. doi: 10.1002/cpa.21595.

[36]

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.

[37]

O.-A. Oleinik and V.-N. Samokhin, Mathematical Models in Boundary Layer Theory, Chapman and Hall/CRC, 1999.

[38]

L. Prandtl, Über flüssigkeits-bewegung bei sehr kleiner reibung, Verhandlungen des III, Internationlen Mathematiker Kongresses, Heidelberg. Teubner, Leipzig, (1904), 484-491.

[39]

L. Rodino, Linear Partial Differential Operators in Gevrey Spaces, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. doi: 10.1142/9789814360036_0002.

[40]

M. Sammartino and R.-E. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. Ⅰ. Existence for Euler and Prandtl equations, Comm. Math. Phys., 192 (1998), 433-461. doi: 10.1007/s002200050304.

[41]

M. Sammartino and R.-E. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. Ⅱ. Construction of the Navier-Stokes solution, Comm. Math. Phys., 192 (1998), 463-491. doi: 10.1007/s002200050305.

[42]

R. Temam and X.-M. Wang, On the behavior of the solutions of the Navier-Stokes equations at vanishing viscosity, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 807-828.

[43]

X.-M. Wang, A Kato type theorem on zero viscosity limit of Navier-Stokes flows, Indiana Univ. Math. J., 50 (2001), 223-241. doi: 10.1512/iumj.2001.50.2098.

[44]

C. WangY.-X. Wang and Z.-F. Zhang, Zero-viscosity limit of the Navier-Stokes equations in the analytic setting, Arch. Ration. Mech. Anal., 224 (2017), 555-595. doi: 10.1007/s00205-017-1083-6.

[45]

S. Wang and Z.-P. Xin, Boundary layer problems in the viscosity-diffusion vanishing limits for the incompressible MHD systems (in Chinese), Sci. China. Math., 47 (2017), 1303-1326. doi: 10.1360/N012016-00211.

[46]

Y.-L. Xiao and Z.-P. Xin, On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition, Comm. Pure Appl. Math., 60 (2007), 1027-1055. doi: 10.1002/cpa.20187.

[47]

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.

[48]

Z.-P. Xin and L.-Q. Zhang, On the global existence of solutions to the Prandtl system, Adv. Math., 181 (2004), 88-133. doi: 10.1016/S0001-8708(03)00046-X.

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J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647

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