August  2019, 24(8): 3819-3841. doi: 10.3934/dcdsb.2018332

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

Received  February 2018 Revised  July 2018 Published  January 2019

Fund Project: The last author is supported by CAPES grant 1579575

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 $
).
Citation: Jens Lorenz, Wilberclay G. Melo, Natã Firmino Rocha. The Magneto–Hydrodynamic equations: Local theory and blow-up of solutions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3819-3841. doi: 10.3934/dcdsb.2018332
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

P. Braz e SilvaW. 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.  Google Scholar

[7]

P. Braz e SilvaW. 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.  Google Scholar

[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. Google Scholar

[9]

M. Cannone, Harmonic Analysis Tools for Solving the Incompressible Navier-Stokes Equations, in: Handbook of Mathematical Fluid Dynamics, Vol. 3, Elsevier, 2004.  Google Scholar

[10]

J.-Y. Chemin, About Navier-Stokes Equations, Publication du Laboratoire Jaques-Louis Lions, Université de Paris Ⅵ R96023, 1996. Google Scholar

[11]

J. Leray, Essai sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Mathematica, 63 (1933), 22-25.   Google Scholar

[12]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

P. Braz e SilvaW. 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.  Google Scholar

[7]

P. Braz e SilvaW. 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.  Google Scholar

[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. Google Scholar

[9]

M. Cannone, Harmonic Analysis Tools for Solving the Incompressible Navier-Stokes Equations, in: Handbook of Mathematical Fluid Dynamics, Vol. 3, Elsevier, 2004.  Google Scholar

[10]

J.-Y. Chemin, About Navier-Stokes Equations, Publication du Laboratoire Jaques-Louis Lions, Université de Paris Ⅵ R96023, 1996. Google Scholar

[11]

J. Leray, Essai sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Mathematica, 63 (1933), 22-25.   Google Scholar

[12]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Igor Kukavica. On regularity for the Navier-Stokes equations in Morrey spaces. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1319-1328. doi: 10.3934/dcds.2010.26.1319

[2]

Chérif Amrouche, Mohamed Meslameni, Šárka Nečasová. Linearized Navier-Stokes equations in $\mathbb{R}^3$: An approach in weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 901-916. doi: 10.3934/dcdss.2014.7.901

[3]

Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637

[4]

Jishan Fan, Yasuhide Fukumoto, Yong Zhou. Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations. Kinetic & Related Models, 2013, 6 (3) : 545-556. doi: 10.3934/krm.2013.6.545

[5]

Yi Zhou, Zhen Lei. Logarithmically improved criteria for Euler and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2715-2719. doi: 10.3934/cpaa.2013.12.2715

[6]

Zijin Li, Xinghong Pan. Some Remarks on regularity criteria of Axially symmetric Navier-Stokes equations. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1333-1350. doi: 10.3934/cpaa.2019064

[7]

Daniel Coutand, Steve Shkoller. Turbulent channel flow in weighted Sobolev spaces using the anisotropic Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations. Communications on Pure & Applied Analysis, 2004, 3 (1) : 1-23. doi: 10.3934/cpaa.2004.3.1

[8]

Chérif Amrouche, María Ángeles Rodríguez-Bellido. On the very weak solution for the Oseen and Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 159-183. doi: 10.3934/dcdss.2010.3.159

[9]

Wenjing Song, Ganshan Yang. The regularization of solution for the coupled Navier-Stokes and Maxwell equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2113-2127. doi: 10.3934/dcdss.2016087

[10]

Daoyuan Fang, Bin Han, Matthias Hieber. Local and global existence results for the Navier-Stokes equations in the rotational framework. Communications on Pure & Applied Analysis, 2015, 14 (2) : 609-622. doi: 10.3934/cpaa.2015.14.609

[11]

Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085

[12]

Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations II: Global existence of small solutions. Evolution Equations & Control Theory, 2012, 1 (1) : 217-234. doi: 10.3934/eect.2012.1.217

[13]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602

[14]

Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149

[15]

Wendong Wang, Liqun Zhang, Zhifei Zhang. On the interior regularity criteria of the 3-D navier-stokes equations involving two velocity components. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2609-2627. doi: 10.3934/dcds.2018110

[16]

Tomoyuki Suzuki. Regularity criteria in weak spaces in terms of the pressure to the MHD equations. Conference Publications, 2011, 2011 (Special) : 1335-1343. doi: 10.3934/proc.2011.2011.1335

[17]

Ming Lu, Yi Du, Zheng-An Yao. Blow-up phenomena for the 3D compressible MHD equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1835-1855. doi: 10.3934/dcds.2012.32.1835

[18]

Ming Lu, Yi Du, Zheng-An Yao, Zujin Zhang. A blow-up criterion for the 3D compressible MHD equations. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1167-1183. doi: 10.3934/cpaa.2012.11.1167

[19]

Minghua Yang, Jinyi Sun. Gevrey regularity and existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1617-1639. doi: 10.3934/cpaa.2017078

[20]

J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (71)
  • HTML views (443)
  • Cited by (0)

[Back to Top]