Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations

José Luiz Boldrini; Jonathan Bravo-Olivares; Eduardo Notte-Cuello; Marko A. Rojas-Medar; José Luiz Boldrini; Jonathan Bravo-Olivares; Eduardo Notte-Cuello; Marko A. Rojas-Medar

doi:10.3934/era.2020091

Electronic Research Archive

2021, Volume 29, Issue 1: 1783-1801. doi: 10.3934/era.2020091

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Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations

1.
Departamento de Matemática, Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, Campinas, SP, Brazil
2.
Departamento de Matemática, Universidad de La Serena, La Serena, Chile
3.
Departamento de Matemática, Universidad de Tarapacá, Casilla 7D, Arica, Chile

Received: 01 January 2020 Revised: 01 June 2020 Published: 16 September 2020
Primary: 35Q30, 35Q35, 76E25, 76D03

We prove some results on the stability of slow stationary solutions of the MHD equations in two- and three-dimensional bounded domains for external force fields that are asymptotically autonomous. Our results show that weak solutions are asymptotically stable in time in the $ L^2 $-norm. Further, assuming certain regularity hypotheses on the problem data, strong solutions are asymptotically stable in the $ H^1 $ and $ H^2 $-norms.
- Magnetohydrodynamics equation,
- stability,
- weak and strong solutions,
- stationary solution,
- evolution solution
Citation: José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations[J]. Electronic Research Archive, 2021, 29(1): 1783-1801. doi: 10.3934/era.2020091

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Abstract

We prove some results on the stability of slow stationary solutions of the MHD equations in two- and three-dimensional bounded domains for external force fields that are asymptotically autonomous. Our results show that weak solutions are asymptotically stable in time in the $ L^2 $-norm. Further, assuming certain regularity hypotheses on the problem data, strong solutions are asymptotically stable in the $ H^1 $ and $ H^2 $-norms.

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