American Institute of Mathematical Sciences

May  2017, 16(3): 745-780. doi: 10.3934/cpaa.2017036

On the decay and stability of global solutions to the 3D inhomogeneous MHD system

 1 School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China 2 Beijing Center for Mathematics and Information Interdisciplinary Sciences, China

Received  March 2016 Revised  January 2017 Published  February 2017

In this paper, we investigative the large time decay and stability to any given global smooth solutions of the 3D incompressible inhomogeneous MHD systems. We prove that given a solution $(a, u, B)$ of (2), the velocity field and the magnetic field decay to zero with an explicit rate, for $u$ which coincide with incompressible inhomogeneous Navier-Stokes equations [1]. In particular, we give the decay rate of higher order derivatives of $u$ and $B$ which are useful to prove our main stability result. For a large solution of (2) denoted by $(a, u, B)$, we show that a small perturbation of the initial data still generates a unique global smooth solution and the smooth solution keeps close to the reference solution $(a, u, B)$. At last, we should mention that the main results in this paper are concerned with large solutions.

Citation: Junxiong Jia, Jigen Peng, Kexue Li. On the decay and stability of global solutions to the 3D inhomogeneous MHD system. Communications on Pure and Applied Analysis, 2017, 16 (3) : 745-780. doi: 10.3934/cpaa.2017036
References:
 [1] H. Abidi, G. Gui and P. Zhang, On the decay and stability of global solutions to the 3 -D inhomogeneous Navier-Stokes equations, Communications on Pure and Applied Mathematics, 64 (2011), 832-881.  doi: 10.1002/cpa.20351. [2] H. Abidi, G. Gui and P. Zhang, On the well-posedness of three-dimensional inhomogeneous Navier-Stokes equations in the critical spaces, Archive for Rational Mechanics and Analysis, 204 (2012), 189-230.  doi: 10.1007/s00205-011-0473-4. [3] H. Abidi and M. Paicu, Global existence for the magnetohydrodynamic system in critical spaces, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 138 (2008), 447-476.  doi: 10.1017/S0308210506001181. [4] S. A. Antontesv, A. V. Kazhikov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, volume 22 of Studies in Mathematics and its Applications, North Holland, 1990. [5] H. Bahouri, J. -Y. Chemin, and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, volume 343 of Grundlehren der mathematischen Wissenschaften. Springer Science & Business Media, 2011. doi: 10.1007/978-3-642-16830-7. [6] C. Cao, D. Regmi and J. Wu, The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion, Journal of Differential Equations, 254 (2013), 2661-2681.  doi: 10.1016/j.jde.2013.01.002. [7] C. Cao and J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Advances in Mathematics, 226 (2011), 1803-1822.  doi: 10.1016/j.aim.2010.08.017. [8] C. Cao, J. Wu and B. Yuan, The 2D incompressible magnetohydrodynamics equations with only magnetic diffusion, SIAM Journal on Mathematical Analysis, 46 (2014), 588-602.  doi: 10.1137/130937718. [9] J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, Journal of Differential Equations, 121 (1995), 314-328.  doi: 10.1006/jdeq.1995.1131. [10] J.-Y. Chemin, M. Paicu and P. Zhang, Global large solutions to 3-D inhomogeneous NavierStokes system with one slow variable, Journal of Differential Equations, 256 (2014), 223-252.  doi: 10.1016/j.jde.2013.09.004. [11] R. Danchin, Local and global well-posedness results for flows of inhomogeneous viscous fluids, Advances in Differential Equations, 9 (2004), 353-386. [12] R. Danchin, Fourier Analysis Methods for PDE's, 2005. [13] R. Danchin, The inviscid limit for density-dependent incompressible fluids, 15 (2006), 637-688. [14] R. Danchin and P. B. Mucha, Incompressible flows with piecewise constant density, Archive for Rational Mechanics and Analysis, 207 (2013), 991-1023.  doi: 10.1007/s00205-012-0586-4. [15] R. Danchin and P. Zhang, Inhomogeneous Navier-Stokes equations in the half-space, with only bounded density, Journal of Functional Analysis, 267 (2014), 2371-2436.  doi: 10.1016/j.jfa.2014.07.017. [16] P. A. Davidson, An Introduction to Magnetohydrodynamics, volume 25 of Cambridge Texts in Applied Mathematics, Cambridge University Press, 2001. doi: 10.1017/CBO9780511626333. [17] G. Duvaut and J.-L. Lions, Inéquations en thermóelasticité et magnétohydrodynamique, Archive for Rational Mechanics and Analysis, 46 (1972), 241-279.  doi: 10.1007/BF00250512. [18] I. Gallagher, D. Iftimie and F. Planchon, Asymptotics and stability for global solutions to the Navier-Stokes equations, 53 (2003), 1387-1424. [19] X. Huang and Y. Wang, Global strong solution to the 2D nonhomogeneous incompressible MHD system, Journal of Differential Equations, 254 (2013), 511-527.  doi: 10.1016/j.jde.2012.08.029. [20] A. V. Kazhikov, Resolution of boundary value problems for nonhomogeneous viscous fluids, Doklady Akademii Nauk, 216 (1974), 1008-1010. [21] G. Ponce, R. Racke, T. C Sideris and E. S Titi, Global stability of large solutions to the 3D Navier-Stokes equations, Communications in Mathematical Physics, 159 (1994), 329-341. [22] P. B. Mucha and R. Danchin, A Lagrangian approach for the incompressible Navier-Stokes equations with variable density, Communications on Pure and Applied Mathematics, 65 (2012), 1458-1480.  doi: 10.1002/cpa.21409. [23] M. E Schonbek, Large time behaviour of solutions to the Navier-Stokes equations, Communications in Partial Differential Equations, 11 (1986), 733-763.  doi: 10.1080/03605308608820443. [24] Carasso S. Alfred. and T. Kato, On subordinated holomorphic semigroups, Trans. Amer. Math. Soc., 327 (1991), 867-878.  doi: 10.2307/2001827.

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References:
 [1] H. Abidi, G. Gui and P. Zhang, On the decay and stability of global solutions to the 3 -D inhomogeneous Navier-Stokes equations, Communications on Pure and Applied Mathematics, 64 (2011), 832-881.  doi: 10.1002/cpa.20351. [2] H. Abidi, G. Gui and P. Zhang, On the well-posedness of three-dimensional inhomogeneous Navier-Stokes equations in the critical spaces, Archive for Rational Mechanics and Analysis, 204 (2012), 189-230.  doi: 10.1007/s00205-011-0473-4. [3] H. Abidi and M. Paicu, Global existence for the magnetohydrodynamic system in critical spaces, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 138 (2008), 447-476.  doi: 10.1017/S0308210506001181. [4] S. A. Antontesv, A. V. Kazhikov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, volume 22 of Studies in Mathematics and its Applications, North Holland, 1990. [5] H. Bahouri, J. -Y. Chemin, and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, volume 343 of Grundlehren der mathematischen Wissenschaften. Springer Science & Business Media, 2011. doi: 10.1007/978-3-642-16830-7. [6] C. Cao, D. Regmi and J. Wu, The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion, Journal of Differential Equations, 254 (2013), 2661-2681.  doi: 10.1016/j.jde.2013.01.002. [7] C. Cao and J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Advances in Mathematics, 226 (2011), 1803-1822.  doi: 10.1016/j.aim.2010.08.017. [8] C. Cao, J. Wu and B. Yuan, The 2D incompressible magnetohydrodynamics equations with only magnetic diffusion, SIAM Journal on Mathematical Analysis, 46 (2014), 588-602.  doi: 10.1137/130937718. [9] J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, Journal of Differential Equations, 121 (1995), 314-328.  doi: 10.1006/jdeq.1995.1131. [10] J.-Y. Chemin, M. Paicu and P. Zhang, Global large solutions to 3-D inhomogeneous NavierStokes system with one slow variable, Journal of Differential Equations, 256 (2014), 223-252.  doi: 10.1016/j.jde.2013.09.004. [11] R. Danchin, Local and global well-posedness results for flows of inhomogeneous viscous fluids, Advances in Differential Equations, 9 (2004), 353-386. [12] R. Danchin, Fourier Analysis Methods for PDE's, 2005. [13] R. Danchin, The inviscid limit for density-dependent incompressible fluids, 15 (2006), 637-688. [14] R. Danchin and P. B. Mucha, Incompressible flows with piecewise constant density, Archive for Rational Mechanics and Analysis, 207 (2013), 991-1023.  doi: 10.1007/s00205-012-0586-4. [15] R. Danchin and P. Zhang, Inhomogeneous Navier-Stokes equations in the half-space, with only bounded density, Journal of Functional Analysis, 267 (2014), 2371-2436.  doi: 10.1016/j.jfa.2014.07.017. [16] P. A. Davidson, An Introduction to Magnetohydrodynamics, volume 25 of Cambridge Texts in Applied Mathematics, Cambridge University Press, 2001. doi: 10.1017/CBO9780511626333. [17] G. Duvaut and J.-L. Lions, Inéquations en thermóelasticité et magnétohydrodynamique, Archive for Rational Mechanics and Analysis, 46 (1972), 241-279.  doi: 10.1007/BF00250512. [18] I. Gallagher, D. Iftimie and F. Planchon, Asymptotics and stability for global solutions to the Navier-Stokes equations, 53 (2003), 1387-1424. [19] X. Huang and Y. Wang, Global strong solution to the 2D nonhomogeneous incompressible MHD system, Journal of Differential Equations, 254 (2013), 511-527.  doi: 10.1016/j.jde.2012.08.029. [20] A. V. Kazhikov, Resolution of boundary value problems for nonhomogeneous viscous fluids, Doklady Akademii Nauk, 216 (1974), 1008-1010. [21] G. Ponce, R. Racke, T. C Sideris and E. S Titi, Global stability of large solutions to the 3D Navier-Stokes equations, Communications in Mathematical Physics, 159 (1994), 329-341. [22] P. B. Mucha and R. Danchin, A Lagrangian approach for the incompressible Navier-Stokes equations with variable density, Communications on Pure and Applied Mathematics, 65 (2012), 1458-1480.  doi: 10.1002/cpa.21409. [23] M. E Schonbek, Large time behaviour of solutions to the Navier-Stokes equations, Communications in Partial Differential Equations, 11 (1986), 733-763.  doi: 10.1080/03605308608820443. [24] Carasso S. Alfred. and T. Kato, On subordinated holomorphic semigroups, Trans. Amer. Math. Soc., 327 (1991), 867-878.  doi: 10.2307/2001827.
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