Advanced Search
Article Contents
Article Contents

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

Abstract Full Text(HTML) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 35Q35; Secondary: 76D03, 76W05.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] H. AbidiG. 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. AbidiG. 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. CaoD. 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. CaoJ. 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. CheminM. 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. PonceR. RackeT. 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.
  • 加载中

Article Metrics

HTML views(1425) PDF downloads(156) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint