### Abstract

In this paper, the properties of weak and strong solutions for the Hall-magnetohydrodynamic system augmented by the effect of elec- tron inertia are studied. First, we establish the existence and uniqueness of local-in-time strong solutions; Then, we prove the existence of global strong solutions under the condition that ∥u _{0} ∥ _{H˙ 1/2} +∥B _{0} ∥ _{H˙ 1/2} +∥∇B _{0} ∥ _{H˙ 1/2} is sufficiently small. Moreover, by applying a cut-off function and gen- eralized energy inequality, we show that the weak solution of electron inertia Hall-MHD system approaches zero as the time t → ∞. Finally, the algebraic decay rate of the weak solution of electron inertia Hall- MHD system is established by using Fourier splitting method and the properties of decay character.

Original language | English |
---|---|

Pages (from-to) | 31-68 |

Number of pages | 38 |

Journal | Advances in Differential Equations |

Volume | 24 |

Issue number | 1-2 |

Publication status | Published - Jan 1 2019 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

*Advances in Differential Equations*,

*24*(1-2), 31-68.

**Well-posedness and large time behavior of solutions for the electron inertial Hall-MHD system.** / Fukumoto, Yasuhide; Zhao, Xiaopeng.

Research output: Contribution to journal › Article

*Advances in Differential Equations*, vol. 24, no. 1-2, pp. 31-68.

}

TY - JOUR

T1 - Well-posedness and large time behavior of solutions for the electron inertial Hall-MHD system

AU - Fukumoto, Yasuhide

AU - Zhao, Xiaopeng

PY - 2019/1/1

Y1 - 2019/1/1

N2 - In this paper, the properties of weak and strong solutions for the Hall-magnetohydrodynamic system augmented by the effect of elec- tron inertia are studied. First, we establish the existence and uniqueness of local-in-time strong solutions; Then, we prove the existence of global strong solutions under the condition that ∥u 0 ∥ H˙ 1/2 +∥B 0 ∥ H˙ 1/2 +∥∇B 0 ∥ H˙ 1/2 is sufficiently small. Moreover, by applying a cut-off function and gen- eralized energy inequality, we show that the weak solution of electron inertia Hall-MHD system approaches zero as the time t → ∞. Finally, the algebraic decay rate of the weak solution of electron inertia Hall- MHD system is established by using Fourier splitting method and the properties of decay character.

AB - In this paper, the properties of weak and strong solutions for the Hall-magnetohydrodynamic system augmented by the effect of elec- tron inertia are studied. First, we establish the existence and uniqueness of local-in-time strong solutions; Then, we prove the existence of global strong solutions under the condition that ∥u 0 ∥ H˙ 1/2 +∥B 0 ∥ H˙ 1/2 +∥∇B 0 ∥ H˙ 1/2 is sufficiently small. Moreover, by applying a cut-off function and gen- eralized energy inequality, we show that the weak solution of electron inertia Hall-MHD system approaches zero as the time t → ∞. Finally, the algebraic decay rate of the weak solution of electron inertia Hall- MHD system is established by using Fourier splitting method and the properties of decay character.

UR - http://www.scopus.com/inward/record.url?scp=85061695990&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85061695990&partnerID=8YFLogxK

M3 - Article

VL - 24

SP - 31

EP - 68

JO - Advances in Differential Equations

JF - Advances in Differential Equations

SN - 1079-9389

IS - 1-2

ER -