Analysis & PDE

  • Anal. PDE
  • Volume 11, Number 4 (2018), 899-918.

Beyond the BKM criterion for the 2D resistive magnetohydrodynamic equations

Léo Agélas

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Abstract

The question of whether the two-dimensional (2D) magnetohydrodynamic (MHD) equations with only magnetic diffusion can develop a finite-time singularity from smooth initial data is a challenging open problem in fluid dynamics and mathematics. In this paper, we derive a regularity criterion less restrictive than the Beale–Kato–Majda (BKM) regularity criterion type, namely any solution ( u , b ) C ( [ 0 , T [ ; H r ( 2 ) ) with r > 2 remains in H r ( 2 ) up to time T under the assumption that

0 T u ( t ) 1 2 log ( e + u ( t ) ) d t < + .

This regularity criterion may stand as a great improvement over the usual BKM regularity criterion, which states that if 0 T × u ( t ) d t < + then the solution ( u , b ) C ( [ 0 , T [ ; H r ( 2 ) ) with r > 2 remains in H r ( 2 ) up to time T . Furthermore, our result applies also to a class of equations arising in hydrodynamics and studied by Elgindi and Masmoudi (2014) for their L ill-posedness.

Article information

Source
Anal. PDE, Volume 11, Number 4 (2018), 899-918.

Dates
Received: 16 January 2017
Revised: 12 September 2017
Accepted: 14 November 2017
First available in Project Euclid: 1 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.apde/1517454158

Digital Object Identifier
doi:10.2140/apde.2018.11.899

Mathematical Reviews number (MathSciNet)
MR3749371

Zentralblatt MATH identifier
1383.35144

Subjects
Primary: 35Q31: Euler equations [See also 76D05, 76D07, 76N10] 35Q61: Maxwell equations

Keywords
MHD Navier–Stokes Euler BKM criterion

Citation

Agélas, Léo. Beyond the BKM criterion for the 2D resistive magnetohydrodynamic equations. Anal. PDE 11 (2018), no. 4, 899--918. doi:10.2140/apde.2018.11.899. https://projecteuclid.org/euclid.apde/1517454158


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References

  • L. Agélas, “Global regularity for logarithmically critical 2D MHD equations with zero viscosity”, Monatsh. Math. 181:2 (2016), 245–266.
  • H. Alfvén, “Existence of electromagnetic-hydrodynamic waves”, Nature 150:3805 (1942), 405–406.
  • H. Bahouri, J.-Y. Chemin, and R. Danchin, Fourier analysis and nonlinear partial differential equations, Grundlehren der Mathematischen Wissenschaften 343, Springer, 2011.
  • J. T. Beale, T. Kato, and A. Majda, “Remarks on the breakdown of smooth solutions for the $3$-D Euler equations”, Comm. Math. Phys. 94:1 (1984), 61–66.
  • D. Biskamp, Nonlinear magnetohydrodynamics, Cambridge Monographs on Plasma Physics 1, Cambridge University Press, 1993.
  • D. Biskamp and E. Schwarz, “On two-dimensional magnetohydrodynamic turbulence”, Phys. Plasmas 8:7 (2001), art. id. 3282.
  • M. E. Brachet, M. D. Bustamante, G. Krstulovic, P. D. Mininni, A. Pouquet, and D. Rosenberg, “Ideal evolution of magnetohydrodynamic turbulence when imposing Taylor–Green symmetries”, Phys. Rev. E 87:1 (2013), art. id. 013110.
  • R. E. Caflisch, I. Klapper, and G. Steele, “Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD”, Comm. Math. Phys. 184:2 (1997), 443–455.
  • C. Cao and J. Wu, “Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion”, Adv. Math. 226:2 (2011), 1803–1822.
  • C. Cao, J. Wu, and B. Yuan, “The 2D incompressible magnetohydrodynamics equations with only magnetic diffusion”, SIAM J. Math. Anal. 46:1 (2014), 588–602.
  • D. Chae, “Nonexistence of self-similar singularities in the viscous magnetohydrodynamics with zero resistivity”, J. Funct. Anal. 254:2 (2008), 441–453.
  • J.-Y. Chemin, Perfect incompressible fluids, Oxford Lecture Series in Mathematics and its Applications 14, Oxford University Press, 1998.
  • Q. Chen, C. Miao, and Z. Zhang, “On the well-posedness of the ideal MHD equations in the Triebel–Lizorkin spaces”, Arch. Ration. Mech. Anal. 195:2 (2010), 561–578.
  • S. I. Chernyshenko, P. Constantin, J. C. Robinson, and E. S. Titi, “A posteriori regularity of the three-dimensional Navier–Stokes equations from numerical computations”, J. Math. Phys. 48:6 (2007), art. id. 065204.
  • T. M. Elgindi and N. Masmoudi, “Ill-posedness results in critical spaces for some equations arising in hydrodynamics”, preprint, 2014.
  • J. Fan, H. Malaikah, S. Monaquel, G. Nakamura, and Y. Zhou, “Global Cauchy problem of 2D generalized MHD equations”, Monatsh. Math. 175:1 (2014), 127–131.
  • S. Gala, A. M. Ragusa, and Z. Ye, “An improved blow-up criterion for smooth solutions of the two-dimensional MHD equations”, Math. Methods Appl. Sci. 40:1 (2017), 279–285.
  • Y. Giga and H. Sohr, “Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier–Stokes equations in exterior domains”, J. Funct. Anal. 102:1 (1991), 72–94.
  • Q. Jiu and D. Niu, “Mathematical results related to a two-dimensional magneto-hydrodynamic equations”, Acta Math. Sci. Ser. B Engl. Ed. 26:4 (2006), 744–756.
  • Q. Jiu and J. Zhao, “A remark on global regularity of 2D generalized magnetohydrodynamic equations”, J. Math. Anal. Appl. 412:1 (2014), 478–484.
  • Q. Jiu and J. Zhao, “Global regularity of 2D generalized MHD equations with magnetic diffusion”, Z. Angew. Math. Phys. 66:3 (2015), 677–687.
  • T. Kato, “Strong $L\sp{p}$–solutions of the Navier–Stokes equation in $\mathbb{R}\sp{m}$, with applications to weak solutions”, Math. Z. 187:4 (1984), 471–480.
  • T. Kato and G. Ponce, “Commutator estimates and the Euler and Navier–Stokes equations”, Comm. Pure Appl. Math. 41:7 (1988), 891–907.
  • R. M. Kerr and A. Brandenburg, “Evidence for a singularity in ideal magnetohydrodynamics: implications for fast reconnection”, Phys. Rev. Lett. 83:6-9 (1999), art. id. 1155.
  • H. Kozono and Y. Taniuchi, “Bilinear estimates and critical Sobolev inequality in BMO, with applications to the Navier–Stokes and the Euler equations”, pp. 39–52 in Mathematical analysis in fluid and gas dynamics (Kyoto, 1999), edited by A. Matsumura and S. Kawashima, Sūrikaisekikenkyūsho Kōkyūroku 1146, RIMS, Kyoto, 2000.
  • Z. Lei and Y. Zhou, “BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity”, Discrete Contin. Dyn. Syst. 25:2 (2009), 575–583.
  • E. Marsch and C.-Y. Tu, “Non-Gaussian probability distributions of solar wind fluctuations”, Ann. Geophysicae 12:12 (1994), 1127–1138.
  • C. S. Ng, A. Bhattacharjee, K. Germaschewski, and S. Galtier, “Anisotropic fluid turbulence in the interstellar medium and solar wind”, Phys. Plasmas 10:5 (2003), 1954–1962.
  • E. R. Priest, Solar magnetohydrodynamics, Springer, 1982.
  • E. Priest and T. Forbes, Magnetic reconnection: MHD theory and applications, Cambridge University Press, 2000.
  • H. Strauss, “Nonlinear three dimensional dynamics of noncircular tokamaks”, Phys. of Fluids 19:1 (1976), 134–140.
  • C. V. Tran, X. Yu, and L. A. K. Blackbourn, “Two-dimensional magnetohydrodynamic turbulence in the limits of infinite and vanishing magnetic Prandtl number”, J. Fluid Mech. 725 (2013), 195–215.
  • C. V. Tran, X. Yu, and Z. Zhai, “On global regularity of 2D generalized magnetohydrodynamic equations”, J. Differential Equations 254:10 (2013), 4194–4216.
  • J. Wu, “Generalized MHD equations”, J. Differential Equations 195:2 (2003), 284–312.
  • J. Wu, “Regularity criteria for the generalized MHD equations”, Comm. Partial Differential Equations 33:1-3 (2008), 285–306.
  • J. Wu, “Global regularity for a class of generalized magnetohydrodynamic equations”, J. Math. Fluid Mech. 13:2 (2011), 295–305.
  • K. Yamazaki, “On the global regularity of two-dimensional generalized magnetohydrodynamics system”, J. Math. Anal. Appl. 416:1 (2014), 99–111.
  • K. Yamazaki, “Remarks on the global regularity of the two-dimensional magnetohydrodynamics system with zero dissipation”, Nonlinear Anal. Theory Methods Appl. 94 (2014), 194–205.
  • Z. Ye and X. Xu, “Global regularity of the two-dimensional incompressible generalized magnetohydrodynamics system”, Nonlinear Anal. Theory Methods Appl. 100 (2014), 86–96.
  • B. Yuan and J. Zhao, “Global Regularity of 2D almost resistive MHD Equations”, Nonlinear Anal. Real World Appl. 41 (2018), 53–65.
  • Y. Zhou and J. Fan, “A regularity criterion for the 2D MHD system with zero magnetic diffusivity”, J. Math. Anal. Appl. 378:1 (2011), 169–172.
  • E. Zweibel and C. Heiles, “Magnetic fields in galaxies and beyond”, Nature 385 (1997), 131–136.