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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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

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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. Google Scholar

[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. Google Scholar

[9]

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

[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. Google Scholar

[11]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

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

[16]

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

[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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[32]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[37]

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

[38]

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

[39]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

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

[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. Google Scholar

[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. Google Scholar

[9]

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

[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. Google Scholar

[11]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

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

[16]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[32]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[37]

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

[38]

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

[39]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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