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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 and 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.

[2]

M. Cannone, A generalization of a theorem by Kato on Navier-Stokes equations, Revista Matemática Iberoamericana, 13 (1997), 515-541.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

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

[2]

M. Cannone, A generalization of a theorem by Kato on Navier-Stokes equations, Revista Matemática Iberoamericana, 13 (1997), 515-541.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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