# American Institute of Mathematical Sciences

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

 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.
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, (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. 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, 74 (2006), 59. Google Scholar [4] C. Cao and J. Wu, Two regularity criteria for the 3D MHD equation,, J. Differential Equations, 248 (2010), 2263. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (1996), 105. 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. 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. 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. Google Scholar [22] M. Sermang and R. Temam, Some mathematical questions related to the MHD equations,, Comm. Pure Appl. Math., 36 (1983), 635. 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. 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. 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. doi: 10.1007/s00332-002-0486-0. Google Scholar

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##### References:
 [1] J. Bergh and J. Löfström, "Interpolation Spaces, An Introduction,", Springer-Verlag, (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. 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, 74 (2006), 59. Google Scholar [4] C. Cao and J. Wu, Two regularity criteria for the 3D MHD equation,, J. Differential Equations, 248 (2010), 2263. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (1996), 105. 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. 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. 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. Google Scholar [22] M. Sermang and R. Temam, Some mathematical questions related to the MHD equations,, Comm. Pure Appl. Math., 36 (1983), 635. 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. 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. 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. doi: 10.1007/s00332-002-0486-0. Google Scholar
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