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 & Applied Analysis, 2017, 16 (3) : 745-780. doi: 10.3934/cpaa.2017036
References:
[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

R. Danchin, Local and global well-posedness results for flows of inhomogeneous viscous fluids, Advances in Differential Equations, 9 (2004), 353-386. Google Scholar

[12]

R. Danchin, Fourier Analysis Methods for PDE's, 2005.Google Scholar

[13]

R. Danchin, The inviscid limit for density-dependent incompressible fluids, 15 (2006), 637-688. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

P. A. Davidson, An Introduction to Magnetohydrodynamics, volume 25 of Cambridge Texts in Applied Mathematics, Cambridge University Press, 2001. doi: 10.1017/CBO9780511626333. Google Scholar

[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. Google Scholar

[18]

I. Gallagher, D. Iftimie and F. Planchon, Asymptotics and stability for global solutions to the Navier-Stokes equations, 53 (2003), 1387-1424. Google Scholar

[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. Google Scholar

[20]

A. V. Kazhikov, Resolution of boundary value problems for nonhomogeneous viscous fluids, Doklady Akademii Nauk, 216 (1974), 1008-1010. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

Carasso S. Alfred. and T. Kato, On subordinated holomorphic semigroups, Trans. Amer. Math. Soc., 327 (1991), 867-878. doi: 10.2307/2001827. Google Scholar

show all references

References:
[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

R. Danchin, Local and global well-posedness results for flows of inhomogeneous viscous fluids, Advances in Differential Equations, 9 (2004), 353-386. Google Scholar

[12]

R. Danchin, Fourier Analysis Methods for PDE's, 2005.Google Scholar

[13]

R. Danchin, The inviscid limit for density-dependent incompressible fluids, 15 (2006), 637-688. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

P. A. Davidson, An Introduction to Magnetohydrodynamics, volume 25 of Cambridge Texts in Applied Mathematics, Cambridge University Press, 2001. doi: 10.1017/CBO9780511626333. Google Scholar

[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. Google Scholar

[18]

I. Gallagher, D. Iftimie and F. Planchon, Asymptotics and stability for global solutions to the Navier-Stokes equations, 53 (2003), 1387-1424. Google Scholar

[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. Google Scholar

[20]

A. V. Kazhikov, Resolution of boundary value problems for nonhomogeneous viscous fluids, Doklady Akademii Nauk, 216 (1974), 1008-1010. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

Carasso S. Alfred. and T. Kato, On subordinated holomorphic semigroups, Trans. Amer. Math. Soc., 327 (1991), 867-878. doi: 10.2307/2001827. Google Scholar

[1]

Fei Chen, Boling Guo, Xiaoping Zhai. Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density. Kinetic & Related Models, 2019, 12 (1) : 37-58. doi: 10.3934/krm.2019002

[2]

Jihong Zhao, Ting Zhang, Qiao Liu. Global well-posedness for the dissipative system modeling electro-hydrodynamics with large vertical velocity component in critical Besov space. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 555-582. doi: 10.3934/dcds.2015.35.555

[3]

Aleksa Srdanov, Radiša Stefanović, Aleksandra Janković, Dragan Milovanović. "Reducing the number of dimensions of the possible solution space" as a method for finding the exact solution of a system with a large number of unknowns. Mathematical Foundations of Computing, 2019, 2 (2) : 83-93. doi: 10.3934/mfc.2019007

[4]

Denis Mercier, Virginie Régnier. Decay rate of the Timoshenko system with one boundary damping. Evolution Equations & Control Theory, 2019, 8 (2) : 423-445. doi: 10.3934/eect.2019021

[5]

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

[6]

Yoshikazu Giga, Yukihiro Seki, Noriaki Umeda. On decay rate of quenching profile at space infinity for axisymmetric mean curvature flow. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1463-1470. doi: 10.3934/dcds.2011.29.1463

[7]

Fei Chen, Yongsheng Li, Huan Xu. Global solution to the 3D nonhomogeneous incompressible MHD equations with some large initial data. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 2945-2967. doi: 10.3934/dcds.2016.36.2945

[8]

Zhuangyi Liu, Ramón Quintanilla. Energy decay rate of a mixed type II and type III thermoelastic system. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1433-1444. doi: 10.3934/dcdsb.2010.14.1433

[9]

Abdelaziz Soufyane, Belkacem Said-Houari. The effect of the wave speeds and the frictional damping terms on the decay rate of the Bresse system. Evolution Equations & Control Theory, 2014, 3 (4) : 713-738. doi: 10.3934/eect.2014.3.713

[10]

Yi-hang Hao, Xian-Gao Liu. The existence and blow-up criterion of liquid crystals system in critical Besov space. Communications on Pure & Applied Analysis, 2014, 13 (1) : 225-236. doi: 10.3934/cpaa.2014.13.225

[11]

Xi Wang, Zuhan Liu, Ling Zhou. Asymptotic decay for the classical solution of the chemotaxis system with fractional Laplacian in high dimensions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 4003-4020. doi: 10.3934/dcdsb.2018121

[12]

Seung-Yeal Ha, Ho Lee, Seok Bae Yun. Uniform $L^p$-stability theory for the space-inhomogeneous Boltzmann equation with external forces. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 115-143. doi: 10.3934/dcds.2009.24.115

[13]

Wei Luo, Zhaoyang Yin. Local well-posedness in the critical Besov space and persistence properties for a three-component Camassa-Holm system with N-peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5047-5066. doi: 10.3934/dcds.2016019

[14]

Zhenhua Guo, Wenchao Dong, Jinjing Liu. Large-time behavior of solution to an inflow problem on the half space for a class of compressible non-Newtonian fluids. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2133-2161. doi: 10.3934/cpaa.2019096

[15]

Hai-Liang Li, Hongjun Yu, Mingying Zhong. Spectrum structure and optimal decay rate of the relativistic Vlasov-Poisson-Landau system. Kinetic & Related Models, 2017, 10 (4) : 1089-1125. doi: 10.3934/krm.2017043

[16]

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

[17]

Ammar Khemmoudj, Yacine Mokhtari. General decay of the solution to a nonlinear viscoelastic modified von-Kármán system with delay. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3839-3866. doi: 10.3934/dcds.2019155

[18]

Xiaopeng Zhao. Space-time decay estimates of solutions to liquid crystal system in $\mathbb{R}^3$. Communications on Pure & Applied Analysis, 2019, 18 (1) : 1-13. doi: 10.3934/cpaa.2019001

[19]

Weike Wang, Xin Xu. Large time behavior of solution for the full compressible navier-stokes-maxwell system. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2283-2313. doi: 10.3934/cpaa.2015.14.2283

[20]

Xiaoping Zhai, Yongsheng Li, Wei Yan. Global well-posedness for the 3-D incompressible MHD equations in the critical Besov spaces. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1865-1884. doi: 10.3934/cpaa.2015.14.1865

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (26)
  • HTML views (90)
  • Cited by (0)

Other articles
by authors

[Back to Top]