American Institute of Mathematical Sciences

Advanced
September  2012, 11(5): 1643-1660. doi: 10.3934/cpaa.2012.11.1643

Regularizing rate estimates for mild solutions of the incompressible Magneto-hydrodynamic system

Qiao Liu 1, and  Shangbin Cui 2,

1. 

Department of Mathematics, Sun Yat-sen University, Guangzhou, Guangdong 510275, China
2. 

Department of Mathematics, Sun Yat-Sen University, Guangzhou, Guangdong 510275

Received  November 2010 Revised  January 2012 Published  March 2012

We establish some regularizing rate estimates for mild solutions of the magneto-hydrodynamic system (MHD). These estimations ensure that there exist positive constants $K_1$ and $K_2$ such that for any $\beta\in\mathbb{Z}^{n}_{+}$ and any $t\in (0,T^\ast)$, where $T^\ast$ is the life-span of the solution, we have $\| (\partial_{x}^{\beta}u(t),\partial_{x}^{\beta}b(t))\|_{q}\leq K_{1}(K_{2}|\beta|)^{|\beta|}t^{-\frac{|\beta|}{2} -\frac{n}{2}(\frac{1}{n}-\frac{1}{q})}$. Spatial analyticity of the solution and temporal decay of global solutions are direct consequences of such estimates.
Keywords: Magneto-hydrodynamic system, regularizing rate, spatial analyticity., mild solution.
Mathematics Subject Classification: 35Q35, 76W05, 35B6.
Citation: Qiao Liu, Shangbin Cui. Regularizing rate estimates for mild solutions of the incompressible Magneto-hydrodynamic system. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1643-1660. doi: 10.3934/cpaa.2012.11.1643
References:
[1]

J. Bergh and J. Löfström, "Interpolation Spaces, An Introduction," Springer-Verlag, New York, 1976.  Google Scholar

[2]

M. Cannone, A generalization of a theorem by Kato on Navier-Stokes equations, Revista Matem$\acutea$tica Iberoamericana, 13 (1997), 515-541.  Google Scholar

[3]

M. Cannone, C. X. Miao, N. Prioux and B. Q. Yuan, The Cauchy problem for the Magneto-hydrodynamic system, self-similar solutions of nonlinear PDE, Banach Center Publications, Institue of Mathematics, Polish Academy of Scuences, Warszawa, 74 (2006), 59-93.  Google Scholar

[4]

C. Cao and J. Wu, Two regularity criteria for the 3D MHD equation, J. Differential Equations, 248 (2010), 2263-2274. doi: 10.1016/j.jde.2009.09.020.  Google Scholar

[5]

S. Cui and C. Guo, Well-posedness of higher-order nonlinear Schrödinger equations in Sobolev spaces $H^s(R^n)$ and applications, Nonlinear Analysis, 67 (2007), 687-707. doi: 10.1016/j.na.2006.06.020.  Google Scholar

[6]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem I, Archive for Rational Mechanics and Analysis, 16 (1964), 269-315. doi: 10.1007/BF00276188.  Google Scholar

[7]

Y. Giga, Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62 (1986), 186-212. Google Scholar

[8]

Y. Giga, K. Inui and S. Matsui, On the Cauchy problem for the Navier Stokes equations with nondecaying initial data, Quaderni di Matematica, 3 (1999), 28-68.  Google Scholar

[9]

Y. Giga and T. Miyakwa, Solutions in $L_r$ of the Navier-Stokes initial value problem, Archive for Rational Mechanics and Analysis, 89 (1985), 267-281. doi: 10.1007/BF00276875.  Google Scholar

[10]

Y. Giga and O. Sawada, On regularizing-decay rate estimates for solutions to the Navier-Stokes initial value problem, Nonlinear Anal. Real World Appl., 1 (2003), 549-562.  Google Scholar

[11]

C. Kahane, On the spatial analyticity of solutions of the Navier-Stokes equations, Archive for Rational Mechanics and Analysis, 33 (1969), 386-405. doi: 10.1007/BF00247697.  Google Scholar

[12]

T. Kato, Strong $L^p$-solutions of the Navier-Stokes equations in $\mathbfR^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182.  Google Scholar

[13]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.  Google Scholar

[14]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Kortweg-de Vries equation via contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620. doi: 10.1002/cpa.3160460405.  Google Scholar

[15]

H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35. doi: 10.1006/aima.2000.1937.  Google Scholar

[16]

H. Kozono and M. Yamazaki, Semilinear heat equations and the navier-stokes equation with distributions in new function spaces as initial data, Comm. Part. Diff. Equ., 19 (1994), 959-1014. doi: 10.1080/03605309408821042.  Google Scholar

[17]

P. G. Lemarié-Rieusset, "Recent Developments in the Navier-Stokes Problem," Chapman and Hall/CRC, 2002.  Google Scholar

[18]

Y. Meyer, Wavelets, paraproducts and Navier-Stokes equations, in "Current Developments in Mathematics (1996)," Cambridge, MA, Int. Press, Boston, MA, (1997), 105-212.  Google Scholar

[19]

C. Miao, B. Yuan and B. Zhang, Well-posedness for the incompressible magneto-hydrodynamic system, Math. Meth. Appl. Sci., 30 (2007), 961-976. doi: 10.1002/mma.820.  Google Scholar

[20]

C. Miao and B. Yuan, On well-posedness of the Cauchy problem for MHD system in Besov spaces, Math. Meth. Appl. Sci., 32 (2009), 53-76. doi: 10.1002/mma.1026.  Google Scholar

[21]

H. Miura and O. Sawada, On the regularizing rate estimates of Koch-Tataru's solution to the Navier-Stokes equations, Asymptotic Analysis, 49 (2006), 1-15.  Google Scholar

[22]

M. Sermang and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. doi: 10.1002/cpa.3160360506.  Google Scholar

[23]

O. Sawada, On analyticity rate estimates of the solutions to the Navier-Stokes equations in Bessel-potential spaces, J. Math. Anal. Appl., 312 (2005), 1-13. doi: 10.1016/j.jmaa.2004.06.068.  Google Scholar

[24]

J. Wu, Regularity results for weak solutions of the 3D MHD equations, Discrete Contin. Dyn. Syst., 10 (2004), 543-556. doi: 10.3934/dcds.2004.10.543.  Google Scholar

[25]

J. Wu, Bounds and new approaches for the 3D MHD equations, J. Nonlinear Sci., 12 (2002), 395-413. doi: 10.1007/s00332-002-0486-0.  Google Scholar

show all references

References:
[1]

J. Bergh and J. Löfström, "Interpolation Spaces, An Introduction," Springer-Verlag, New York, 1976.  Google Scholar

[2]

M. Cannone, A generalization of a theorem by Kato on Navier-Stokes equations, Revista Matem$\acutea$tica Iberoamericana, 13 (1997), 515-541.  Google Scholar

[3]

M. Cannone, C. X. Miao, N. Prioux and B. Q. Yuan, The Cauchy problem for the Magneto-hydrodynamic system, self-similar solutions of nonlinear PDE, Banach Center Publications, Institue of Mathematics, Polish Academy of Scuences, Warszawa, 74 (2006), 59-93.  Google Scholar

[4]

C. Cao and J. Wu, Two regularity criteria for the 3D MHD equation, J. Differential Equations, 248 (2010), 2263-2274. doi: 10.1016/j.jde.2009.09.020.  Google Scholar

[5]

S. Cui and C. Guo, Well-posedness of higher-order nonlinear Schrödinger equations in Sobolev spaces $H^s(R^n)$ and applications, Nonlinear Analysis, 67 (2007), 687-707. doi: 10.1016/j.na.2006.06.020.  Google Scholar

[6]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem I, Archive for Rational Mechanics and Analysis, 16 (1964), 269-315. doi: 10.1007/BF00276188.  Google Scholar

[7]

Y. Giga, Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62 (1986), 186-212. Google Scholar

[8]

Y. Giga, K. Inui and S. Matsui, On the Cauchy problem for the Navier Stokes equations with nondecaying initial data, Quaderni di Matematica, 3 (1999), 28-68.  Google Scholar

[9]

Y. Giga and T. Miyakwa, Solutions in $L_r$ of the Navier-Stokes initial value problem, Archive for Rational Mechanics and Analysis, 89 (1985), 267-281. doi: 10.1007/BF00276875.  Google Scholar

[10]

Y. Giga and O. Sawada, On regularizing-decay rate estimates for solutions to the Navier-Stokes initial value problem, Nonlinear Anal. Real World Appl., 1 (2003), 549-562.  Google Scholar

[11]

C. Kahane, On the spatial analyticity of solutions of the Navier-Stokes equations, Archive for Rational Mechanics and Analysis, 33 (1969), 386-405. doi: 10.1007/BF00247697.  Google Scholar

[12]

T. Kato, Strong $L^p$-solutions of the Navier-Stokes equations in $\mathbfR^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182.  Google Scholar

[13]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.  Google Scholar

[14]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Kortweg-de Vries equation via contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620. doi: 10.1002/cpa.3160460405.  Google Scholar

[15]

H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35. doi: 10.1006/aima.2000.1937.  Google Scholar

[16]

H. Kozono and M. Yamazaki, Semilinear heat equations and the navier-stokes equation with distributions in new function spaces as initial data, Comm. Part. Diff. Equ., 19 (1994), 959-1014. doi: 10.1080/03605309408821042.  Google Scholar

[17]

P. G. Lemarié-Rieusset, "Recent Developments in the Navier-Stokes Problem," Chapman and Hall/CRC, 2002.  Google Scholar

[18]

Y. Meyer, Wavelets, paraproducts and Navier-Stokes equations, in "Current Developments in Mathematics (1996)," Cambridge, MA, Int. Press, Boston, MA, (1997), 105-212.  Google Scholar

[19]

C. Miao, B. Yuan and B. Zhang, Well-posedness for the incompressible magneto-hydrodynamic system, Math. Meth. Appl. Sci., 30 (2007), 961-976. doi: 10.1002/mma.820.  Google Scholar

[20]

C. Miao and B. Yuan, On well-posedness of the Cauchy problem for MHD system in Besov spaces, Math. Meth. Appl. Sci., 32 (2009), 53-76. doi: 10.1002/mma.1026.  Google Scholar

[21]

H. Miura and O. Sawada, On the regularizing rate estimates of Koch-Tataru's solution to the Navier-Stokes equations, Asymptotic Analysis, 49 (2006), 1-15.  Google Scholar

[22]

M. Sermang and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. doi: 10.1002/cpa.3160360506.  Google Scholar

[23]

O. Sawada, On analyticity rate estimates of the solutions to the Navier-Stokes equations in Bessel-potential spaces, J. Math. Anal. Appl., 312 (2005), 1-13. doi: 10.1016/j.jmaa.2004.06.068.  Google Scholar

[24]

J. Wu, Regularity results for weak solutions of the 3D MHD equations, Discrete Contin. Dyn. Syst., 10 (2004), 543-556. doi: 10.3934/dcds.2004.10.543.  Google Scholar

[25]

J. Wu, Bounds and new approaches for the 3D MHD equations, J. Nonlinear Sci., 12 (2002), 395-413. doi: 10.1007/s00332-002-0486-0.  Google Scholar

[1]

Xuhui Peng, Jianhua Huang, Yan Zheng. Exponential mixing for the fractional Magneto-Hydrodynamic equations with degenerate stochastic forcing. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4479-4506. doi: 10.3934/cpaa.2020204
[2]

Paul Deuring. Spatial asymptotics of mild solutions to the time-dependent Oseen system. Communications on Pure & Applied Analysis, 2021, 20 (5) : 1833-1849. doi: 10.3934/cpaa.2021044
[3]

Haifeng Hu, Kaijun Zhang. Stability of the stationary solution of the cauchy problem to a semiconductor full hydrodynamic model with recombination-generation rate. Kinetic & Related Models, 2015, 8 (1) : 117-151. doi: 10.3934/krm.2015.8.117
[4]

Yumi Yahagi. Construction of unique mild solution and continuity of solution for the small initial data to 1-D Keller-Segel system. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021099
[5]

Jens Lorenz, Wilberclay G. Melo, Natã Firmino Rocha. The Magneto–Hydrodynamic equations: Local theory and blow-up of solutions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3819-3841. doi: 10.3934/dcdsb.2018332
[6]

Jaeseop Ahn, Jimyeong Kim, Ihyeok Seo. On the radius of spatial analyticity for defocusing nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 423-439. doi: 10.3934/dcds.2020016
[7]

Boling Guo, Guangwu Wang. Existence of the solution for the viscous bipolar quantum hydrodynamic model. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3183-3210. doi: 10.3934/dcds.2017136
[8]

Daniele Davino, Ciro Visone. Rate-independent memory in magneto-elastic materials. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 649-691. doi: 10.3934/dcdss.2015.8.649
[9]

Shu-Guang Shao, Shu Wang, Wen-Qing Xu, Yu-Li Ge. On the local C1, α solution of ideal magneto-hydrodynamical equations. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 2103-2113. doi: 10.3934/dcds.2017090
[10]

Shijin Ding, Changyou Wang, Huanyao Wen. Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 357-371. doi: 10.3934/dcdsb.2011.15.357
[11]

A. Alexandrou Himonas, Gerson Petronilho. A $ G^{\delta, 1} $ almost conservation law for mCH and the evolution of its radius of spatial analyticity. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2031-2050. doi: 10.3934/dcds.2020351
[12]

Marion Acheritogaray, Pierre Degond, Amic Frouvelle, Jian-Guo Liu. Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system. Kinetic & Related Models, 2011, 4 (4) : 901-918. doi: 10.3934/krm.2011.4.901
[13]

Jihong Zhao, Qiao Liu, Shangbin Cui. Global existence and stability for a hydrodynamic system in the nematic liquid crystal flows. Communications on Pure & Applied Analysis, 2013, 12 (1) : 341-357. doi: 10.3934/cpaa.2013.12.341
[14]

Peter Markowich, Jesús Sierra. Non-uniqueness of weak solutions of the Quantum-Hydrodynamic system. Kinetic & Related Models, 2019, 12 (2) : 347-356. doi: 10.3934/krm.2019015
[15]

Hyun-Jung Kim. Stochastic parabolic Anderson model with time-homogeneous generalized potential: Mild formulation of solution. Communications on Pure & Applied Analysis, 2019, 18 (2) : 795-807. doi: 10.3934/cpaa.2019038
[16]

Qiaoyi Hu, Zhijun Qiao. Analyticity, Gevrey regularity and unique continuation for an integrable multi-component peakon system with an arbitrary polynomial function. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6975-7000. doi: 10.3934/dcds.2016103
[17]

Jing Zhang. The analyticity and exponential decay of a Stokes-wave coupling system with viscoelastic damping in the variational framework. Evolution Equations & Control Theory, 2017, 6 (1) : 135-154. doi: 10.3934/eect.2017008
[18]

Youshan Tao, Michael Winkler. A chemotaxis-haptotaxis system with haptoattractant remodeling: Boundedness enforced by mild saturation of signal production. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2047-2067. doi: 10.3934/cpaa.2019092
[19]

Yongming Liu, Lei Yao. Global solution and decay rate for a reduced gravity two and a half layer model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2613-2638. doi: 10.3934/dcdsb.2018267
[20]

Kazuo Yamazaki. Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2193-2207. doi: 10.3934/dcds.2015.35.2193

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy