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November  2019, 18(6): 3035-3057. doi: 10.3934/cpaa.2019136

Asymptotic behavior of solutions to incompressible electron inertial Hall-MHD system in $ \mathbb{R}^3 $

Ning Duan 1,2, , Yasuhide Fukumoto 3, and  Xiaopeng Zhao 1,2,,

1. 

School of Science, Jiangnan University, Wuxi 214122, China
2. 

School of Science, Northeastern University, Shenyang 110819, China
3. 

Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan

* Corresponding author

Received  September 2018 Revised  September 2018 Published  May 2019

In this paper, by using Fourier splitting method and the properties of decay character $ r^* $, we consider the decay rate on higher order derivative of solutions to 3D incompressible electron inertial Hall-MHD system in Sobolev space $ H^s(\mathbb{R}^3)\times H^{s+1}(\mathbb{R}^3) $ for $ s\in\mathbb{N}^+ $. Moreover, based on a parabolic interpolation inequality, bootstrap argument and some weighted estimates, we also address the space-time decay properties of strong solutions in $ \mathbb{R}^3 $.

Keywords: Electron inertial Hall-MHD system, decay character, decay rate, weighted decay.
Mathematics Subject Classification: Primary: 35Q35, 35B40; Secondary: 76W05.
Citation: Ning Duan, Yasuhide Fukumoto, Xiaopeng Zhao. Asymptotic behavior of solutions to incompressible electron inertial Hall-MHD system in $ \mathbb{R}^3 $. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3035-3057. doi: 10.3934/cpaa.2019136
References:
[1]

H. M. Abdelhamid, Y. Kawazura and Z. Yoshida, Hamiltonian formalism of extended magnetohydrodynamics, J. Phys. A: Math. Theor., 48 (2015), 235502.  doi: 10.1088/1751-8113/48/23/235502.  Google Scholar

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C. Bjorland and M. E. Schonbek, Poincaré's inequality and diffusive evolution equations, Adv. Differential Equations, 14 (2009), 241-260.   Google Scholar

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L. Brandolese, Characterization of solutions to dissipative systems with sharp algebraic decay, SIAM J. Math. Anal., 48 (2016), 1616-1633.  doi: 10.1137/15M1040475.  Google Scholar

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L. Brandolese and M. E. Schonbek, Large time decay and growth for solutions of a viscous Boussinesq system, Trans. Amer. Math. Soc., 364 (2012), 5057-5090.  doi: 10.1090/S0002-9947-2012-05432-8.  Google Scholar

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D. Chae and M. E. Schonbek, On the temporal decay for the Hall-magnetohydrodynamic equations, J. Differential Equations, 255 (2013), 3971-3982.  doi: 10.1016/j.jde.2013.07.059.  Google Scholar

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Y. Fukumoto and X. Zhao, Well-posedness and large time behavior of solutions for the electron inertial Hall-MHD system, Adv. Differential Equations, 24 (2019), 31-68.   Google Scholar

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Q. Jiu and H. Yu, Decay of solutions to the three-dimensional generalized Navier-Stokes equations, Asymptotic Anal., 94 (2015), 105-124.  doi: 10.3233/ASY-151307.  Google Scholar

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I. Kukavica and J. J. Torres, Weighted $L^p$ decay for solutions of the Navier-Stokes equations, Comm. Partial Differential Equations, 32 (2007), 819-831.  doi: 10.1080/03605300600781659.  Google Scholar

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T. Miyakawa, On space-time decay properties of nonstationary incompressible Navier-Stokes flows in $\mathbb{R}^n$, Funkcial. Ekvac., 43 (2000), 541-557.   Google Scholar

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C. J. Niche and M. E. Schonbek, Decay characterization of solutions to dissipative equations, J. London Math. Soc., 91 (2015), 573-595.  doi: 10.1112/jlms/jdu085.  Google Scholar

[24]

C. J. Niche, Decay characterization of solutions to Navier-Stokes-Voigt equations in terms of the initial datum, J. Differential Equations, 260 (2016), 4440-4453.  doi: 10.1016/j.jde.2015.11.014.  Google Scholar

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M. E. Schonbek, $L^2$ decay for weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 88 (1985), 209-222.  doi: 10.1007/BF00752111.  Google Scholar

[26]

M. E. Schonbek, Large time behaviour of solutions to the Navier-Stokes equations, Comm. Partial Differential Equations, 11 (1986), 733-763.  doi: 10.1080/03605308608820443.  Google Scholar

[27]

M. Schonbek and T. Schonbek, On the boundedness and decay of moments of solutions to the Navier-Stokes equations, Adv. Differential Equations, 5 (2000), 861-898.   Google Scholar

[28]

S. Takahashi, A weighted equation approach to decay rate estimates for the Navier-Stokes equations, Nonlinear Anal., 37 (1999), 751-789.  doi: 10.1016/S0362-546X(98)00070-4.  Google Scholar

[29]

S. Weng, Space-time decay estimates for the incompressible viscous resistive MHD and Hall-MHD equations, J. Funct. Anal., 270 (2016), 2168-2187.  doi: 10.1016/j.jfa.2016.01.021.  Google Scholar

[30]

S. Weng, Remarks on asymptotic behaviors of strong solutions to a viscous Boussinesq system, Math. Methods Appl. Sci., 39 (2016), 4398-4418.  doi: 10.1002/mma.3868.  Google Scholar

[31]

S. Weng, On analyticity and temporal decay rates of solutions to the viscous resistive Hall-MHD system, J. Differential Equations, 260 (2016), 6504-6524.  doi: 10.1016/j.jde.2016.01.003.  Google Scholar

[32]

X. Zhao, Decay of solutions to a new Hall-MHD system in $\mathbb{R}^3$, C. R. Acad. Sci. Paris, Ser. I., 355 (2017), 310-317.  doi: 10.1016/j.crma.2017.01.019.  Google Scholar

[33]

X. Zhao, Space-time decay estimates of solutions to Liquid crystal system in $\mathbb{R}^3$, Commun. Pure Anal. Appl., 18 (2019), 1-13.  doi: 10.3934/cpaa.2019001.  Google Scholar

[34]

X. Zhao and M. Zhu, Global well-posedness and asymptotic behavior of solutions for the three-dimensional MHD equations with Hall and ion-slip effects, Z. Angew. Math. Phys., 69 (2018), Art. 22, 13 pp. doi: 10.1007/s00033-018-0907-z.  Google Scholar

show all references

References:
[1]

H. M. Abdelhamid, Y. Kawazura and Z. Yoshida, Hamiltonian formalism of extended magnetohydrodynamics, J. Phys. A: Math. Theor., 48 (2015), 235502.  doi: 10.1088/1751-8113/48/23/235502.  Google Scholar

[2]

N. Andrés, L. Martin, P. Dmitruk and D. Gómez, Effects of electron inertia in collisionless magnetic reconnection, Phy. Plasmas, 21 (2014), 072904.   Google Scholar

[3]

N. Andrés, C. Gonzalez, L. Martin, P. Dmitruk and D. Gómez, Two-fluid turbulence including electron inertia, Phy. Plasmas, 21 (2014), 122305.   Google Scholar

[4]

N. Andrés, P. Dmitruk and D. Gómez, Influence of the Hall effect and electron inertia in collisionless magnetic reconnection, Phy. Plasmas, 23 (2016), 022903.   Google Scholar

[5]

C. T. Anh and P. T. Trang, Decay characterization of solutions to the viscous Camassa-Holm equations, Nonlinearity, 31 (2018), 621-650.  doi: 10.1088/1361-6544/aa96ce.  Google Scholar

[6]

C. Bjorland and M. E. Schonbek, Poincaré's inequality and diffusive evolution equations, Adv. Differential Equations, 14 (2009), 241-260.   Google Scholar

[7]

L. Brandolese, Characterization of solutions to dissipative systems with sharp algebraic decay, SIAM J. Math. Anal., 48 (2016), 1616-1633.  doi: 10.1137/15M1040475.  Google Scholar

[8]

L. Brandolese and M. E. Schonbek, Large time decay and growth for solutions of a viscous Boussinesq system, Trans. Amer. Math. Soc., 364 (2012), 5057-5090.  doi: 10.1090/S0002-9947-2012-05432-8.  Google Scholar

[9]

L. Brandolese, On a non-solenoidal approximation to the incompressible Navier-Stokes equations, J. Lond. Math. Soc. (2), 96 (2017), 326-344.  doi: 10.1112/jlms.12063.  Google Scholar

[10]

L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compos. Math., 53 (1984), 259-275.   Google Scholar

[11]

D. Chae and M. E. Schonbek, On the temporal decay for the Hall-magnetohydrodynamic equations, J. Differential Equations, 255 (2013), 3971-3982.  doi: 10.1016/j.jde.2013.07.059.  Google Scholar

[12] J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511526404.  Google Scholar
[13]

M. Dai and M. E. Schonbek, Asymptotic behavior of solutions to the liquid crystal system in $H^m(\mathbb{R}^3)$, SIAM J. Math. Anal., 46 (2014), 3131-3150.  doi: 10.1137/120895342.  Google Scholar

[14]

Y. Fukumoto and X. Zhao, Well-posedness and large time behavior of solutions for the electron inertial Hall-MHD system, Adv. Differential Equations, 24 (2019), 31-68.   Google Scholar

[15]

Q. Jiu and H. Yu, Decay of solutions to the three-dimensional generalized Navier-Stokes equations, Asymptotic Anal., 94 (2015), 105-124.  doi: 10.3233/ASY-151307.  Google Scholar

[16]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Commun. Pure. Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.  Google Scholar

[17]

K. Kimura and P. J. Morrison, On energy conservation in extended magnetohydrodynamics, Phy. Plasmas, 21 (2014), 082101.   Google Scholar

[18]

I. Kukavica, Space-time decay for solutions of the Navier-Stokes equations, Indiana Univ. Math. J., 50 (2001), 205-222.  doi: 10.1512/iumj.2001.50.2084.  Google Scholar

[19]

I. Kukavica, On the weighted decay for solutions of the Navier-Stokes system, Nonlinear Anal., 70 (2009), 2466-2470.  doi: 10.1016/j.na.2008.03.031.  Google Scholar

[20]

I. Kukavica and J. J. Torres, Weighted bounds for the velocity and the vorticity for the Navier-Stokes equations, Nonlinearity, 19 (2006), 293-303.  doi: 10.1088/0951-7715/19/2/003.  Google Scholar

[21]

I. Kukavica and J. J. Torres, Weighted $L^p$ decay for solutions of the Navier-Stokes equations, Comm. Partial Differential Equations, 32 (2007), 819-831.  doi: 10.1080/03605300600781659.  Google Scholar

[22]

T. Miyakawa, On space-time decay properties of nonstationary incompressible Navier-Stokes flows in $\mathbb{R}^n$, Funkcial. Ekvac., 43 (2000), 541-557.   Google Scholar

[23]

C. J. Niche and M. E. Schonbek, Decay characterization of solutions to dissipative equations, J. London Math. Soc., 91 (2015), 573-595.  doi: 10.1112/jlms/jdu085.  Google Scholar

[24]

C. J. Niche, Decay characterization of solutions to Navier-Stokes-Voigt equations in terms of the initial datum, J. Differential Equations, 260 (2016), 4440-4453.  doi: 10.1016/j.jde.2015.11.014.  Google Scholar

[25]

M. E. Schonbek, $L^2$ decay for weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 88 (1985), 209-222.  doi: 10.1007/BF00752111.  Google Scholar

[26]

M. E. Schonbek, Large time behaviour of solutions to the Navier-Stokes equations, Comm. Partial Differential Equations, 11 (1986), 733-763.  doi: 10.1080/03605308608820443.  Google Scholar

[27]

M. Schonbek and T. Schonbek, On the boundedness and decay of moments of solutions to the Navier-Stokes equations, Adv. Differential Equations, 5 (2000), 861-898.   Google Scholar

[28]

S. Takahashi, A weighted equation approach to decay rate estimates for the Navier-Stokes equations, Nonlinear Anal., 37 (1999), 751-789.  doi: 10.1016/S0362-546X(98)00070-4.  Google Scholar

[29]

S. Weng, Space-time decay estimates for the incompressible viscous resistive MHD and Hall-MHD equations, J. Funct. Anal., 270 (2016), 2168-2187.  doi: 10.1016/j.jfa.2016.01.021.  Google Scholar

[30]

S. Weng, Remarks on asymptotic behaviors of strong solutions to a viscous Boussinesq system, Math. Methods Appl. Sci., 39 (2016), 4398-4418.  doi: 10.1002/mma.3868.  Google Scholar

[31]

S. Weng, On analyticity and temporal decay rates of solutions to the viscous resistive Hall-MHD system, J. Differential Equations, 260 (2016), 6504-6524.  doi: 10.1016/j.jde.2016.01.003.  Google Scholar

[32]

X. Zhao, Decay of solutions to a new Hall-MHD system in $\mathbb{R}^3$, C. R. Acad. Sci. Paris, Ser. I., 355 (2017), 310-317.  doi: 10.1016/j.crma.2017.01.019.  Google Scholar

[33]

X. Zhao, Space-time decay estimates of solutions to Liquid crystal system in $\mathbb{R}^3$, Commun. Pure Anal. Appl., 18 (2019), 1-13.  doi: 10.3934/cpaa.2019001.  Google Scholar

[34]

X. Zhao and M. Zhu, Global well-posedness and asymptotic behavior of solutions for the three-dimensional MHD equations with Hall and ion-slip effects, Z. Angew. Math. Phys., 69 (2018), Art. 22, 13 pp. doi: 10.1007/s00033-018-0907-z.  Google Scholar

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