Study of the Blow Up of the Maximal Solution to the Three-Dimensional Magnetohydrodynamic System in Lei-Lin-Gevrey Spaces

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Ridha Selmi, Jamel Benameur, Study of the Blow Up of the Maximal Solution to the Three-Dimensional Magnetohydrodynamic System in Lei-Lin-Gevrey Spaces, Int. J. Anal. Appl., 18 (3) (2020), 421-438.

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Ridha Selmi, Jamel Benameur

Abstract

In this paper, we investigate existence, uniqueness and blowup in finite time of the local solution to the three dimensional magnetohydrodynamic system, in Gevrey-Lei-Lin spaces. To prove the blowup results and give the blow profile as a function of time, two key points are used. The first is a frequency decomposition of the spectrum of the initial data. This allows to use Leray theory. The second is a technical lemma we proved to state that the Lei-Lin space is an interpolation space between the Gevrey-Lei-Lin and the Lebesgue square integrable functions spaces. To prove uniqueness, we use a penalization procedure and energy methods. About existence, we split the initial condition into low frequencies part and high frequencies part. The former are considered as initial data to the linear part of the system. The latter will be taken as small as needed, so that smallness theory applies and allows to run a fixed point argument.

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References

  1. J. T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Commun. Math. Phys. 94 (1984), 61””66.
  2. J. Benameur, On the blow-up criterion of 3D Navier-Stokes equations, J. Math. Anal. Appl. 371 (2010), 719--727.
  3. J. Benameur, On the blow-up criterion of the periodic incompressible fluids, Math. Methods Appl. Sci. 36 (2) (2013), 143””153.
  4. J. Benameur, On the exponential type explosion of Navier-Stokes equations, Nonlinear Anal., Theory Methods Appl. 103 (2014), 87-97.
  5. J. Benameur and L. Jlali, On the blow-up criterion of 3D-NSE in Sobolev-Gevrey spaces, J. Math. Fluid Mech. 18 (4) (2016), 805-822.
  6. J. Benameur and R. Selmi, Anisotropic Rotating MHD System in Critical Anisotropic Spaces, Mem. Differential Equations Math. Phys. 44 (2008), 23-44.
  7. J. Benameur and R. Selmi, Study of anisotropic MHD system in anisotropic Sobolev spaces, Ann. Fac. Sci. Toulouse Math., 17 (1) (2008), 1-22.
  8. J. Benameur and R. Selmi, Long time decay to the Leray solution of the two-dimensional Navier-Stokes equations, Bull. London Math. Soc. 44 (5) (2012), 1001-1019.
  9. J. Benameur and R. Selmi, Time decay and exponential stability of solutions to the periodic 3D Navier-Stokes equation in critical spaces, Math. Methods Appl. Sci. 37 (17) (2014), 2817-2828.
  10. M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations, Handbook of Mathematical Fluid Dynamics, vol 3, eds. S. Friedlander and D. Serre, Elsevier, 2003.
  11. J-Y. Chemin, Remarques sur l'existence global pour le systeme de Navier-Stokes incompressible, SIAM J. Math. Anal. 23 (1) (1992), 27-50.
  12. H. Fujita and T. Kato, On the Navier-Stokes initial value problem I, Arch. Ration. Mech. Anal. 16 (1964), 269-315.
  13. T. Kato, Strong Lp solutions of the Navier-Stokes equations in Rm with applications to weak solutions, Math. Z. 187 (4)(1984), 471-480.
  14. H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math. 157 (1) (2001), 22-35.
  15. Z. Lei and F. Lin, Global mild solutions of Navier-Stokes equations, Commun. Pure Appl. Math. 64 (2011), 1297-1304.
  16. Z. Lei and L. Xiuting, The local well-posedness, blow-up criteria and Gevrey regularity of solutions for a two-component high-order Camassa-Holm system, Nonlinear Anal., Real. World Appl. 35 (2017), 414-440.
  17. J. Leray, Essai sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63 (1933), 22-25.
  18. R. Selmi, Convergence results for MHD system, Internat. J. Math. Math. Sci. 2006 (2006), Article ID 28704.
  19. R. Selmi, Asymptotic study of mixed rotating MHD system, Bull. Korean Math. Soc. 47 (2) (2010), 231-250.
  20. R. Selmi, Global well-posedness and convergence results for 3D-regularized Boussinesq system, Canad. J. Math. 64 (6) (2012), 1415-1435.