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When fast diffusion and reactive growth both induce accelerating invasions
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 |
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 $.
References:
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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. |
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