The Magneto–Hydrodynamic equations: Local theory and blow-up of solutions

Jens Lorenz; Wilberclay G. Melo; Natã Firmino Rocha

doi:10.3934/dcdsb.2018332

Article Contents

2019, Volume 24, Issue 8: 3819-3841. Doi: 10.3934/dcdsb.2018332

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The Magneto–Hydrodynamic equations: Local theory and blow-up of solutions

1.
Department of Mathematics and Statistics, The University of New Mexico, Albuquerque NM 87131, United States of America
2.
Departamento de Matematica, Universidade Federal de Sergipe, Sao Cristovao SE 49100-000, Brasil
3.
Departamento de Matematica, Universidade Federal de Minas Gerais, Belo Horizonte MG 31270-901, Brasil

^* Corresponding author: Jens Lorenz
^* Corresponding author: Jens Lorenz

Received: February 1, 2018

Revised: July 1, 2018

Early access: January 4, 2019

Published online: January 4, 2019

The last author is supported by CAPES grant 1579575.

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Abstract

This work establishes local existence and uniqueness as well as blow-up criteria for solutions $ (u,b)(x,t) $ of the Magneto–Hydrodynamic equations in Sobolev–Gevrey spaces $ \dot{H}^s_{a,\sigma}(\mathbb{R}^3) $. More precisely, we prove that there is a time $ T>0 $ such that $ (u,b)\in C([0,T];\dot{H}_{a,\sigma}^s(\mathbb{R}^3)) $ for $ a>0, \sigma\geq1 $ and $ \frac{1}{2}<s<\frac{3}{2} $. If the maximal time interval of existence is finite, $ 0\leq t < T^* $, then the blow–up inequality

$ \frac{C_1\exp\{C_2(T^*-t)^{-\frac{1}{3\sigma}}\}\;\;\;\;\;\;\;}{\;\;\;\;(T^*-t)^{q}\;\;\;\;} \;\;\;\;\;\;\;\;\;\;\leq \|(u,b)(t)\|_{\dot{H}_{a,\sigma}^s(\mathbb{R}^3)} \quad \mbox{with}\,\, q = {\frac{2(s\sigma+\sigma_0)+1}{6\sigma}} $

holds for $ 0\leq t<T^*, \frac{1}{2}<s<\frac{3}{2} $, $ a>0 $, $ \sigma> 1 $ ($ 2\sigma_0 $ is the integer part of $ 2\sigma $).

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Mathematics Subject Classification: Primary: 35B44, 35Q30, 76D03, 76D05, 76W05.

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References

[1]	J. Benameur, On the blow-up criterion of 3D Navier-Stokes equations, J. Math. Anal. Appl., 371 (2010), 719-727. doi: 10.1016/j.jmaa.2010.06.007.
[2]	J. Benameur, On the blow-up criterion of the periodic incompressible fluids, Math. Methods Appl. Sci., 36 (2013), 143-153. doi: 10.1002/mma.2577.
[3]	J. Benameur, On the exponential type explosion of Navier-Stokes equations, Nonlinear Analysis, 103 (2014), 87-97. doi: 10.1016/j.na.2014.03.011.
[4]	J. Benameur and L. Jlali, On the blow-up criterion of 3D-NSE in Sobolev-Gevrey spaces, J. Math. Fluid Mech., 18 (2016), 805-822. doi: 10.1007/s00021-016-0263-8.
[5]	A. Biswas, Local existence and Gevrey regularity of 3-D Navier-Stokes equations with lp initial data, J. Differential Equations, 215 (2005), 429-447. doi: 10.1016/j.jde.2004.12.012.
[6]	P. Braz e Silva, W. G. Melo and P. R. Zingano, Some remarks on the paper "On the blow up criterion of 3D Navier-Stokes equations" by J. Benameur, C. R. Acad. Sci. Paris, Ser. I, 352 (2014), 913-915. doi: 10.1016/j.crma.2014.09.012.
[7]	P. Braz e Silva, W. G. Melo and P. R. Zingano, Lower bounds on blow-up of solutions for magneto-micropolar fluid systems in homogeneous sobolev spaces, Acta Appl Math., 147 (2017), 1-17. doi: 10.1007/s10440-016-0065-2.
[8]	P. Braz e Silva, W. G. Melo and N. F. Rocha, Existence, Uniqueness and Blow-up of Solutions for the 3D Navier-Stokes Equations in Homogeneous Sobolev-Gevrey Spaces, Submitted, 2018.
[9]	M. Cannone, Harmonic Analysis Tools for Solving the Incompressible Navier-Stokes Equations, in: Handbook of Mathematical Fluid Dynamics, Vol. 3, Elsevier, 2004.
[10]	J.-Y. Chemin, About Navier-Stokes Equations, Publication du Laboratoire Jaques-Louis Lions, Université de Paris Ⅵ R96023, 1996.
[11]	J. Leray, Essai sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Mathematica, 63 (1933), 22-25.
[12]	J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354.
[13]	J. Lorenz and P. R. Zingano, Properties at potential blow-up times for the incompressible Navier-Stokes equations, Bol. Soc. Paran. Mat., 35 (2017), 127-158. doi: 10.5269/bspm.v35i2.27508.
[14]	W. G. Melo, The magneto-micropolar equations with periodic boundary conditions: Solution properties at potential blow-up times, J. Math. Anal. Appl., 435 (2016), 1194-1209. doi: 10.1016/j.jmaa.2015.11.005.
[15]	J. Yuan, Existence theorem and blow-up criterion of the strong solutions to the magneto-micropolar fluid equations, Math. Methods Appl. Sci., 31 (2008), 1113-1130. doi: 10.1002/mma.967.