September  2015, 14(5): 1865-1884. doi: 10.3934/cpaa.2015.14.1865

Global well-posedness for the 3-D incompressible MHD equations in the critical Besov spaces

1. 

School of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China

2. 

Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640

3. 

College of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan 453007

Received  September 2014 Revised  February 2015 Published  June 2015

In this paper, we consider the global well-posedness of the incompressible magnetohydrodynamic equations with initial data $(u_0,b_0)$ in the critical Besov space $\dot{B}_{2,1}^{1/2}(\mathbb{R}^3)\times \dot{B}_{2,1}^{1/2}(\mathbb{R}^3)$. Compared with [30], making full use of the algebraical structure of the equations, we relax the smallness condition in the third component of the initial velocity field and magnetic field. More precisely, we prove that there exist two positive constants $\varepsilon_0$ and $C_0$ such that if \begin{eqnarray} (\|u_0^h\|_{\dot{B}_{2,1}^{1/2}} +\|b_0^h\|_{\dot{B}_{2,1}^{1/2}}) \exp\{C_0(\frac{1}{\mu}+\frac{1}{\nu})^3 (\|u_0^3\|_{\dot{B}_{2,1}^{1/2}} +\|b_0^3\|_{\dot{B}_{2,1}^{1/2}})^2\} \le \varepsilon_0\mu\nu, \end{eqnarray} then the 3-D incompressible magnetohydrodynamic system has a unique global solution $(u,b)\in C([0,+\infty);\dot{B}_{2,1}^{1/2})\cap L^1((0,+\infty);\dot{B}_{2,1}^{5/2})\times C([0,+\infty);\dot{B}_{2,1}^{1/2})\cap L^1((0,+\infty);\dot{B}_{2,1}^{5/2}).$ Finally, we analyze the long behavior of the solution and get some decay estimates which imply that for any $t>0$ the solution $(u(t),b(t))\in C^{\infty}(\mathbb{R}^3)\times C^{\infty}(\mathbb{R}^3)$.
Citation: Xiaoping Zhai, Yongsheng Li, Wei Yan. Global well-posedness for the 3-D incompressible MHD equations in the critical Besov spaces. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1865-1884. doi: 10.3934/cpaa.2015.14.1865
References:
[1]

H. Abidi and M. Paicu, Existence globale pour un fluide inhomogéne,, \emph{Ann. Inst. Fourier (Grenoble)}, 57 (2007), 883.

[2]

H. Abidi and M. Paicu, Global existence for the magnetohydrodynamic system in critical spaces,, \emph{Proc. Roy. Soc. Edinburgh Sect. A.}, 138 (2008), 447. doi: 10.1017/S0308210506001181.

[3]

J. M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires,, \emph{Ann. Sci. \'Ecole Norm. Sup.}, 14 (1981), 209.

[4]

R. Caflisch, I. Klapper and G. Steele, Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD,, \emph{Comm. Math. Phys.}, 184 (1997), 443. doi: 10.1007/s002200050067.

[5]

M. Cannone, Ondelettes Paraproduits et Navier-Stokes,, Diderot editeur, (1995).

[6]

C. Cao, D. Regmi and J. Wu, The 2-D MHD equations with horizontal dissipation and horizontal magnetic diffusion,, \emph{J. Diff. Eqns.}, 254 (2013), 2661. doi: 10.1016/j.jde.2013.01.002.

[7]

C. Cao and J. Wu, Global regularity for the 2-D MHD equations with mixed partial dissipation and magnetic diffusion,, \emph{Adv. Math.}, 226 (2011), 1803. doi: 10.1016/j.aim.2010.08.017.

[8]

J. Y. Chemin, Remarques sur l'existence globale pour le système de Navier-Stokes incompressible,, \emph{SIAM Journal on Mathematical Analysis}, 23 (1992), 20. doi: 10.1137/0523002.

[9]

J. Y. Chemin, Théorèmes d'unicité pour le système de Navier-Stokes tridimensionnel,, \emph{J. Anal. Math.}, 77 (1999), 27. doi: 10.1007/BF02791256.

[10]

J. Y. Chemin and I. Gallagher, On the global wellposedness of the 3-D Navier-Stokes equations with large initial data,, \emph{Ann. Sci. \'Ecole Norm. Sup.}, 39 (2006), 679. doi: 10.1016/j.ansens.2006.07.002.

[11]

Q. Chen and C. Miao, Global well-posedness for the micropolar fluid system in critical Besov spaces,, \emph{J. Diff. Eqns.}, 252 (2012), 2698. doi: 10.1016/j.jde.2011.09.035.

[12]

Q. Chen, C. Miao and Z. Zhang, On the regularity criterion of weak solution for the 3-D viscous magneto-hydrodynamics equations,, \emph{Comm. Math. Phys.}, 284 (2008), 919. doi: 10.1007/s00220-008-0545-y.

[13]

G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique,, \emph{Arch. Ration. Mech. Anal.}, 46 (1972), 241.

[14]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem I,, \emph{Arch. Ration. Mech. Anal.}, 16 (1964), 269.

[15]

C. He and X. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations,, \emph{J. Funct. Anal.}, 227 (2005), 113. doi: 10.1016/j.jfa.2005.06.009.

[16]

C. He and X. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations,, \emph{J. Diff. Eqns.}, 213 (2005), 235. doi: 10.1016/j.jde.2004.07.002.

[17]

D. Iftimie, The 3-D Navier-Stokes equations seen as a perturbation of the 2-D Navier-Stokes equations,, \emph{Bull. Soc. Math. France}, 127 (1999), 473.

[18]

D. Iftimie, The resolution of the Navier-Stokes equations in anisotropic spaces,, \emph{Rev. Mat. Iberoamericana}, 15 (1999), 1. doi: 10.4171/RMI/248.

[19]

D. Iftimie and G. R. Raugel, Some results on the Navier-Stokes equations in thin 3-D domains,, \emph{J. Diff. Eqns.}, 169 (2001), 281. doi: 10.1006/jdeq.2000.3900.

[20]

T. Kato, Strong $L^q$ solutions of the Navier-Stokes equations in $\mathbbR^n$ with applications to weak solutions,, \emph{Mathematische Zeitschrift}, 187 (1984), 471. doi: 10.1007/BF01174182.

[21]

I. Kukavica, W. Rusin and M. Ziane, A class of large $BMO^{-1}$ non-oscillatory data for the Navier-Stokes equations,, \emph{J. Math. Fluid Mech.}, 16 (2014), 293. doi: 10.1007/s00021-013-0160-3.

[22]

I. Kukavica, W. Rusin and M. Ziane, A class of solutions of the Navier-Stokes equations with large data,, \emph{J. Diff Eqns.}, 255 (2013), 1492. doi: 10.1016/j.jde.2013.05.009.

[23]

I. Kukavica and M. Ziane, Regularity of the Navier-Stokes equation in a thin periodic domain with large data,, \emph{Discrete Contin. Dyn. Syst.}, 16 (2006), 67. doi: 10.3934/dcds.2006.16.67.

[24]

H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations,, \emph{Adv. Math.}, 157 (2001), 22. doi: 10.1006/aima.2000.1937.

[25]

H. Kozono, Weak and classical solutions of the two-dimensional magnetohydrodynamic equations,, \emph{Tohoku Mathematical Journal.}, 41 (1989), 471. doi: 10.2748/tmj/1178227774.

[26]

Z. Lei and Y. Zhou, BKM's criterion and global weak solutions for magnetohydrody-namics with zero viscosity equations,, \emph{Discrete Contin. Dyn. Syst-A}, 25 (2009), 575. doi: 10.3934/dcds.2009.25.575.

[27]

PG. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem,, Chapman $&$ Hall/CRC: London, (2002). doi: 10.1201/9781420035674.

[28]

F. Lin and P. Zhang, Global small solutions to MHD type system (I): 3-D case,, \emph{Comm. Pure Appl. Math.}, ().

[29]

F. Lin, L. Xu and P. Zhang, Global small solutions to 2-D incompressible MHD system,, arXiv:1302.5877v2 [math.AP]., ().

[30]

C. Miao and B. Yuan, On the well-posedness of the Cauchy problem for an MHD system in Besov spaces,, \emph{Math. Meth. Appl. Sci.}, 32 (2009), 53. doi: 10.1002/mma.1026.

[31]

M. Paicu and P. Zhang, Global solutions to the 3-D incompressible anisotropic Navier-Stokes system in the critical spaces,, \emph{Comm. Math. Phys.}, 307 (2011), 713. doi: 10.1007/s00220-011-1350-6.

[32]

M. Paicu and P. Zhang, Global solutions to the 3-D incompressible inhomogeneous Navier-Stokes system,, \emph{J. Funct. Anal.}, 262 (2012), 3556. doi: 10.1016/j.jfa.2012.01.022.

[33]

J. Peetre, New Thoughts on Besov Spaces,, Duke Univers. Math. Ser., (1976).

[34]

F. Planchon, Global strong solutions in Sobolev or Lebesgue spaces to the incompressible Navier-Stokes equations in $\mathbbR^3$,, \emph{Annales lInstitut Henri Poincar{\'e}}, 13 (1996), 319.

[35]

G. Raugel and G. R. Sell, Équations de Navier-Stokes dans des domaines minces en dimension trois: régularité globale,, \emph{C. R. Acad. Sci. Paris S{\'e}r. I Math.}, 309 (1989), 299.

[36]

X. Ren, J. Wu, Z. Xiang and Z. Zhang, Global existence and decay of smooth solution for the 2-D MHD equations without magentic diffusion,, \emph{J. Funct. Anal.}, 267 (2014), 503. doi: 10.1016/j.jfa.2014.04.020.

[37]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations,, \emph{Comm. Pure Appl. Math.}, 36 (1983), 635. doi: 10.1002/cpa.3160360506.

[38]

J. Wu, Y. Wu and X. Xu, Global small solution to the 2-D MHD system with a velocity damping term,, arXiv:1311.6185v1 [math.AP]., ().

[39]

T. Zhang, An elementary proof of the global existence and uniqueness theorem to 2-D incompressible non-resistive MHD system,, arXiv:1404.5681v1 [math.AP]., ().

[40]

Y. Zhou, Remarks on regularities for the 3-D MHD equations,, \emph{Discrete Contin. Dyn. Syst.}, 12 (2005), 881. doi: 10.3934/dcds.2005.12.881.

show all references

References:
[1]

H. Abidi and M. Paicu, Existence globale pour un fluide inhomogéne,, \emph{Ann. Inst. Fourier (Grenoble)}, 57 (2007), 883.

[2]

H. Abidi and M. Paicu, Global existence for the magnetohydrodynamic system in critical spaces,, \emph{Proc. Roy. Soc. Edinburgh Sect. A.}, 138 (2008), 447. doi: 10.1017/S0308210506001181.

[3]

J. M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires,, \emph{Ann. Sci. \'Ecole Norm. Sup.}, 14 (1981), 209.

[4]

R. Caflisch, I. Klapper and G. Steele, Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD,, \emph{Comm. Math. Phys.}, 184 (1997), 443. doi: 10.1007/s002200050067.

[5]

M. Cannone, Ondelettes Paraproduits et Navier-Stokes,, Diderot editeur, (1995).

[6]

C. Cao, D. Regmi and J. Wu, The 2-D MHD equations with horizontal dissipation and horizontal magnetic diffusion,, \emph{J. Diff. Eqns.}, 254 (2013), 2661. doi: 10.1016/j.jde.2013.01.002.

[7]

C. Cao and J. Wu, Global regularity for the 2-D MHD equations with mixed partial dissipation and magnetic diffusion,, \emph{Adv. Math.}, 226 (2011), 1803. doi: 10.1016/j.aim.2010.08.017.

[8]

J. Y. Chemin, Remarques sur l'existence globale pour le système de Navier-Stokes incompressible,, \emph{SIAM Journal on Mathematical Analysis}, 23 (1992), 20. doi: 10.1137/0523002.

[9]

J. Y. Chemin, Théorèmes d'unicité pour le système de Navier-Stokes tridimensionnel,, \emph{J. Anal. Math.}, 77 (1999), 27. doi: 10.1007/BF02791256.

[10]

J. Y. Chemin and I. Gallagher, On the global wellposedness of the 3-D Navier-Stokes equations with large initial data,, \emph{Ann. Sci. \'Ecole Norm. Sup.}, 39 (2006), 679. doi: 10.1016/j.ansens.2006.07.002.

[11]

Q. Chen and C. Miao, Global well-posedness for the micropolar fluid system in critical Besov spaces,, \emph{J. Diff. Eqns.}, 252 (2012), 2698. doi: 10.1016/j.jde.2011.09.035.

[12]

Q. Chen, C. Miao and Z. Zhang, On the regularity criterion of weak solution for the 3-D viscous magneto-hydrodynamics equations,, \emph{Comm. Math. Phys.}, 284 (2008), 919. doi: 10.1007/s00220-008-0545-y.

[13]

G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique,, \emph{Arch. Ration. Mech. Anal.}, 46 (1972), 241.

[14]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem I,, \emph{Arch. Ration. Mech. Anal.}, 16 (1964), 269.

[15]

C. He and X. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations,, \emph{J. Funct. Anal.}, 227 (2005), 113. doi: 10.1016/j.jfa.2005.06.009.

[16]

C. He and X. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations,, \emph{J. Diff. Eqns.}, 213 (2005), 235. doi: 10.1016/j.jde.2004.07.002.

[17]

D. Iftimie, The 3-D Navier-Stokes equations seen as a perturbation of the 2-D Navier-Stokes equations,, \emph{Bull. Soc. Math. France}, 127 (1999), 473.

[18]

D. Iftimie, The resolution of the Navier-Stokes equations in anisotropic spaces,, \emph{Rev. Mat. Iberoamericana}, 15 (1999), 1. doi: 10.4171/RMI/248.

[19]

D. Iftimie and G. R. Raugel, Some results on the Navier-Stokes equations in thin 3-D domains,, \emph{J. Diff. Eqns.}, 169 (2001), 281. doi: 10.1006/jdeq.2000.3900.

[20]

T. Kato, Strong $L^q$ solutions of the Navier-Stokes equations in $\mathbbR^n$ with applications to weak solutions,, \emph{Mathematische Zeitschrift}, 187 (1984), 471. doi: 10.1007/BF01174182.

[21]

I. Kukavica, W. Rusin and M. Ziane, A class of large $BMO^{-1}$ non-oscillatory data for the Navier-Stokes equations,, \emph{J. Math. Fluid Mech.}, 16 (2014), 293. doi: 10.1007/s00021-013-0160-3.

[22]

I. Kukavica, W. Rusin and M. Ziane, A class of solutions of the Navier-Stokes equations with large data,, \emph{J. Diff Eqns.}, 255 (2013), 1492. doi: 10.1016/j.jde.2013.05.009.

[23]

I. Kukavica and M. Ziane, Regularity of the Navier-Stokes equation in a thin periodic domain with large data,, \emph{Discrete Contin. Dyn. Syst.}, 16 (2006), 67. doi: 10.3934/dcds.2006.16.67.

[24]

H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations,, \emph{Adv. Math.}, 157 (2001), 22. doi: 10.1006/aima.2000.1937.

[25]

H. Kozono, Weak and classical solutions of the two-dimensional magnetohydrodynamic equations,, \emph{Tohoku Mathematical Journal.}, 41 (1989), 471. doi: 10.2748/tmj/1178227774.

[26]

Z. Lei and Y. Zhou, BKM's criterion and global weak solutions for magnetohydrody-namics with zero viscosity equations,, \emph{Discrete Contin. Dyn. Syst-A}, 25 (2009), 575. doi: 10.3934/dcds.2009.25.575.

[27]

PG. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem,, Chapman $&$ Hall/CRC: London, (2002). doi: 10.1201/9781420035674.

[28]

F. Lin and P. Zhang, Global small solutions to MHD type system (I): 3-D case,, \emph{Comm. Pure Appl. Math.}, ().

[29]

F. Lin, L. Xu and P. Zhang, Global small solutions to 2-D incompressible MHD system,, arXiv:1302.5877v2 [math.AP]., ().

[30]

C. Miao and B. Yuan, On the well-posedness of the Cauchy problem for an MHD system in Besov spaces,, \emph{Math. Meth. Appl. Sci.}, 32 (2009), 53. doi: 10.1002/mma.1026.

[31]

M. Paicu and P. Zhang, Global solutions to the 3-D incompressible anisotropic Navier-Stokes system in the critical spaces,, \emph{Comm. Math. Phys.}, 307 (2011), 713. doi: 10.1007/s00220-011-1350-6.

[32]

M. Paicu and P. Zhang, Global solutions to the 3-D incompressible inhomogeneous Navier-Stokes system,, \emph{J. Funct. Anal.}, 262 (2012), 3556. doi: 10.1016/j.jfa.2012.01.022.

[33]

J. Peetre, New Thoughts on Besov Spaces,, Duke Univers. Math. Ser., (1976).

[34]

F. Planchon, Global strong solutions in Sobolev or Lebesgue spaces to the incompressible Navier-Stokes equations in $\mathbbR^3$,, \emph{Annales lInstitut Henri Poincar{\'e}}, 13 (1996), 319.

[35]

G. Raugel and G. R. Sell, Équations de Navier-Stokes dans des domaines minces en dimension trois: régularité globale,, \emph{C. R. Acad. Sci. Paris S{\'e}r. I Math.}, 309 (1989), 299.

[36]

X. Ren, J. Wu, Z. Xiang and Z. Zhang, Global existence and decay of smooth solution for the 2-D MHD equations without magentic diffusion,, \emph{J. Funct. Anal.}, 267 (2014), 503. doi: 10.1016/j.jfa.2014.04.020.

[37]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations,, \emph{Comm. Pure Appl. Math.}, 36 (1983), 635. doi: 10.1002/cpa.3160360506.

[38]

J. Wu, Y. Wu and X. Xu, Global small solution to the 2-D MHD system with a velocity damping term,, arXiv:1311.6185v1 [math.AP]., ().

[39]

T. Zhang, An elementary proof of the global existence and uniqueness theorem to 2-D incompressible non-resistive MHD system,, arXiv:1404.5681v1 [math.AP]., ().

[40]

Y. Zhou, Remarks on regularities for the 3-D MHD equations,, \emph{Discrete Contin. Dyn. Syst.}, 12 (2005), 881. doi: 10.3934/dcds.2005.12.881.

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