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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 $

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 $.

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. AbdelhamidY. 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. AndrésL. MartinP. Dmitruk and D. Gómez, Effects of electron inertia in collisionless magnetic reconnection, Phy. Plasmas, 21 (2014), 072904. Google Scholar

[3]

N. AndrésC. GonzalezL. MartinP. Dmitruk and D. Gómez, Two-fluid turbulence including electron inertia, Phy. Plasmas, 21 (2014), 122305. Google Scholar

[4]

N. AndrésP. 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. CaffarelliR. 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.
[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

show all references

References:
[1]

H. M. AbdelhamidY. 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. AndrésL. MartinP. Dmitruk and D. Gómez, Effects of electron inertia in collisionless magnetic reconnection, Phy. Plasmas, 21 (2014), 072904. Google Scholar

[3]

N. AndrésC. GonzalezL. MartinP. Dmitruk and D. Gómez, Two-fluid turbulence including electron inertia, Phy. Plasmas, 21 (2014), 122305. Google Scholar

[4]

N. AndrésP. 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. CaffarelliR. 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.
[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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