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

Ning Duan; Yasuhide Fukumoto; Xiaopeng Zhao

doi:10.3934/cpaa.2019136

Previous Article
Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity
CPAA Home
This Issue
Next Article
When fast diffusion and reactive growth both induce accelerating invasions

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 and 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.
[2]	N. Andrés, L. Martin, P. Dmitruk and D. Gómez, Effects of electron inertia in collisionless magnetic reconnection, Phy. Plasmas, 21 (2014), 072904.
[3]	N. Andrés, C. Gonzalez, L. Martin, P. Dmitruk and D. Gómez, Two-fluid turbulence including electron inertia, Phy. Plasmas, 21 (2014), 122305.
[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.
[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.
[6]	C. Bjorland and M. E. Schonbek, Poincaré's inequality and diffusive evolution equations, Adv. Differential Equations, 14 (2009), 241-260.
[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.
[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.
[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.
[10]	L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compos. Math., 53 (1984), 259-275.
[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.
[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.
[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.
[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.
[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.
[17]	K. Kimura and P. J. Morrison, On energy conservation in extended magnetohydrodynamics, Phy. Plasmas, 21 (2014), 082101.
[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.
[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.
[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.
[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.
[22]	T. Miyakawa, On space-time decay properties of nonstationary incompressible Navier-Stokes flows in $\mathbb{R}^n$, Funkcial. Ekvac., 43 (2000), 541-557.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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.
[2]	N. Andrés, L. Martin, P. Dmitruk and D. Gómez, Effects of electron inertia in collisionless magnetic reconnection, Phy. Plasmas, 21 (2014), 072904.
[3]	N. Andrés, C. Gonzalez, L. Martin, P. Dmitruk and D. Gómez, Two-fluid turbulence including electron inertia, Phy. Plasmas, 21 (2014), 122305.
[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.
[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.
[6]	C. Bjorland and M. E. Schonbek, Poincaré's inequality and diffusive evolution equations, Adv. Differential Equations, 14 (2009), 241-260.
[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.
[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.
[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.
[10]	L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compos. Math., 53 (1984), 259-275.
[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.
[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.
[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.
[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.
[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.
[17]	K. Kimura and P. J. Morrison, On energy conservation in extended magnetohydrodynamics, Phy. Plasmas, 21 (2014), 082101.
[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.
[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.
[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.
[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.
[22]	T. Miyakawa, On space-time decay properties of nonstationary incompressible Navier-Stokes flows in $\mathbb{R}^n$, Funkcial. Ekvac., 43 (2000), 541-557.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

[1]	Jincheng Gao, Zheng-An Yao. Global existence and optimal decay rates of solutions for compressible Hall-MHD equations. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3077-3106. doi: 10.3934/dcds.2016.36.3077
[2]	Yanxia Niu, Yinxia Wang, Qingnian Zhang. Decay rate of global solutions to three dimensional generalized MHD system. Evolution Equations and Control Theory, 2021, 10 (2) : 249-258. doi: 10.3934/eect.2020064
[3]	Lvqiao Liu. On the global existence to Hall-MHD system. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022044
[4]	Jishan Fan, Fucai Li, Gen Nakamura. Low Mach number limit of the full compressible Hall-MHD system. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1731-1740. doi: 10.3934/cpaa.2017084
[5]	Denis Mercier, Virginie Régnier. Decay rate of the Timoshenko system with one boundary damping. Evolution Equations and Control Theory, 2019, 8 (2) : 423-445. doi: 10.3934/eect.2019021
[6]	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
[7]	Zhuangyi Liu, Ramón Quintanilla. Energy decay rate of a mixed type II and type III thermoelastic system. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1433-1444. doi: 10.3934/dcdsb.2010.14.1433
[8]	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 and Control Theory, 2014, 3 (4) : 713-738. doi: 10.3934/eect.2014.3.713
[9]	Mohammed Aassila. On energy decay rate for linear damped systems. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 851-864. doi: 10.3934/dcds.2002.8.851
[10]	Bopeng Rao. Optimal energy decay rate in a damped Rayleigh beam. Discrete and Continuous Dynamical Systems, 1998, 4 (4) : 721-734. doi: 10.3934/dcds.1998.4.721
[11]	Monica Conti, Lorenzo Liverani, Vittorino Pata. On the optimal decay rate of the weakly damped wave equation. Communications on Pure and Applied Analysis, 2022, 21 (10) : 3421-3424. doi: 10.3934/cpaa.2022107
[12]	Hai-Liang Li, Hongjun Yu, Mingying Zhong. Spectrum structure and optimal decay rate of the relativistic Vlasov-Poisson-Landau system. Kinetic and Related Models, 2017, 10 (4) : 1089-1125. doi: 10.3934/krm.2017043
[13]	Xing Wu, Keqin Su. Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6057-6068. doi: 10.3934/dcdsb.2021002
[14]	Eugenii Shustin. Exponential decay of oscillations in a multidimensional delay differential system. Conference Publications, 2003, 2003 (Special) : 809-816. doi: 10.3934/proc.2003.2003.809
[15]	Tariel Sanikidze, A.F. Tedeev. On the temporal decay estimates for the degenerate parabolic system. Communications on Pure and Applied Analysis, 2013, 12 (4) : 1755-1768. doi: 10.3934/cpaa.2013.12.1755
[16]	Frank Jochmann. Decay of the polarization field in a Maxwell Bloch system. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 663-676. doi: 10.3934/dcds.2003.9.663
[17]	Abderrahmane Youkana, Salim A. Messaoudi. General and optimal decay for a quasilinear parabolic viscoelastic system. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1307-1316. doi: 10.3934/dcdss.2021129
[18]	Yongming Liu, Lei Yao. Global solution and decay rate for a reduced gravity two and a half layer model. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2613-2638. doi: 10.3934/dcdsb.2018267
[19]	Yoshikazu Giga, Yukihiro Seki, Noriaki Umeda. On decay rate of quenching profile at space infinity for axisymmetric mean curvature flow. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1463-1470. doi: 10.3934/dcds.2011.29.1463
[20]	Yuan Xu, Fujun Zhou, Weihua Gong. Global Well-posedness and Optimal Decay Rate of the Quasi-static Incompressible Navier–Stokes–Fourier–Maxwell–Poisson System. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1537-1565. doi: 10.3934/cpaa.2022028

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors