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July  2021, 41(7): 3045-3062. doi: 10.3934/dcds.2020397

The Littlewood-Paley $ pth $-order moments in three-dimensional MHD turbulence

Yao Nie and  Jia Yuan ,

School of Mathematics and Systems Science, Beihang University, Beijing 100191, China

* Corresponding author: Jia Yuan

Received  July 2020 Revised  October 2020 Published  July 2021 Early access  December 2020

Fund Project: The work is supported by NSF grant No.11871087 and No.11771423. The first author is supported by the Academic Excellence Foundation of BUAA for PhD students and China Scholarship Council No.201906020100

In this paper, we consider the Littlewood-Paley $ p $th-order ($ 1\le p<\infty $) moments of the three-dimensional MHD periodic equations, which are defined by the infinite-time and space average of $ L^p $-norm of velocity and magnetic fields involved in the spectral cut-off operator $ \dot\Delta_m $. Our results imply that in some cases, $ k^{-\frac{1}{3}} $ is an upper bound at length scale $ 1/k $. This coincides with the scaling law of many observations on astrophysical systems and simulations in terms of 3D MHD turbulence.

Keywords: Incompressible MHD equations, three-dimensional, Littlewood-Paley, Kolmogorov, turbulence.
Mathematics Subject Classification: Primary: 35Q35; Secondary: 42B37, 76D03, 76W05.
Citation: Yao Nie, Jia Yuan. The Littlewood-Paley $ pth $-order moments in three-dimensional MHD turbulence. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3045-3062. doi: 10.3934/dcds.2020397
References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, 343, Springer-Verlag, 2011. doi: 10.1007%2F978-3-642-16830-7.  Google Scholar

[2]

A. Basu and J. K. Bhattacharjee, Universal properties of three-dimensional magnetohydrodynamic turbulence: do Alfvén waves matter?, J. Stat. Mech., 2005 (2005), P07002. doi: 10.1088/1742-5468/2005/07/P07002.  Google Scholar

[3]

A. Basu, A. Sain, S. K. Dhar and R. Pandit, Multiscaling in models of magnetohydrodynamic turbulence, Phys. Rev. Lett., 81 (1998), 2687-2690.  doi: 10.1103/PhysRevLett.81.2687.  Google Scholar

[4]

D. Biskamp and W-C. Müller, Scaling properties of three-dimensional isotropic magnetohydrodynamic turbulence, Phys. Plasmas, 7 (2000), 4889-4900.  doi: 10.1063/1.1322562.  Google Scholar

[5]

M. Cannone, Harmonic analysis tools for solving incompressible Navier-Stokes equations, Handbook of Mathmatical Fluid Dynamics vol 3,161–244, North-Holland, Amsterdam, 2004.  Google Scholar

[6]

Q. Chen, C. Miao and Z. Zhang, A new Bernstein's inequality and the 2D dissipative quasi-geostrophic equation, Comm. Math. Phys., 271 (2007), 821-838.  doi: 10.1007/s00220-007-0193-7.  Google Scholar

[7]

Q. Chen, C. Miao and Z. Zhang, On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations, Comm. Math. Phys., 284 (2008), 919-930.  doi: 10.1007/s00220-008-0545-y.  Google Scholar

[8]

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 (2010), 561-578.  doi: 10.1007/s00205-008-0213-6.  Google Scholar

[9]

J. Cho, E. T. Vishniac and A. Lazarian, Simulations of magnetohydrodynamic turbulence in a strongly magnetized medium, Astrophys. J., 564 (2002), 291-301.  doi: 10.1086/324186.  Google Scholar

[10]

J. Cho and E. T. Vishniac, The anisotropy of magnetohydrodynamic Alfvénic turbulence, Astrophys. J., 539 (2000), 273-282.  doi: 10.1086/309213.  Google Scholar

[11]

P. Constantin, Euler equations, Navier-Stokes equations and turbulence, in Mathematical Foundation of Turbulent Viscous Flows, Lecture Notes in Math. Vol. 1871, Berlin: Springer, 2006, 1–43. doi: 10.1007%2F11545989_1.  Google Scholar

[12]

P. Constantin, The Littlewood-Paley spectrum in two-dimensional turbulence, Theor. Comput. Fluid Dyn., 9 (1997), 183-189.  doi: 10.1007/s001620050039.  Google Scholar

[13]

E. Falgarone and T. Passot, Turbulence and Magnetic Fields in Astrophysics, Lecture Notes in Physics, Springer, 2003. doi: 10.1007%2F3-540-36238-X.  Google Scholar

[14]

Y. Gupta, B. J. Rickett and W. A. Coles, Refractive interstellar scintillation of pulsar intensities at 74 MHz, Astrophysical J., 403 (1993), 183-201.  doi: 10.1086/172193.  Google Scholar

[15]

E. Hopf, Über die anfangswertaufgabe für die hydrodynamischen grundgleichungen, Math. Nachr., 4 (1951), 213-231.  doi: 10.1002/mana.3210040121.  Google Scholar

[16]

P. S. Iroshnikov, Turbulence of a conducting fluid in a strong magnetic field, Soviet Astronom. AJ, 7 (1964), 566-571.   Google Scholar

[17]

A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Proceedings of the Royal Society A, 434 (1991), 9-13.  doi: 10.1098/rspa.1991.0075.  Google Scholar

[18]

A. N. Kolmogorov, Dissipation of energy in the locally isotropic turbulence, Proceedings of the Royal Society A, 434 (1991), 15-17.  doi: 10.1098/rspa.1991.0076.  Google Scholar

[19]

R. H. Kraichnan, Lagrangian-history closure approximation for turbulence, Phys. Fluids, 8 (1965), 575-598.  doi: 10.1063/1.1761271.  Google Scholar

[20]

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

[21] C. Miao, J. Wu and Z. Zhang, Littlewood-Paley Theory and Applications to Fluid Dynamics Equations, Monographs on Modern pure mathematics, No. 142, Beijing: Science Press, 2012.   Google Scholar
[22]

W-C. Müller and D. Biskamp, Scaling properties of three-dimensional magnetohydrodynamic turbulence, Phys. Rev. Lett., 84 (2000), 475-478.  doi: 10.1103/PhysRevLett.84.475.  Google Scholar

[23]

F. Otto and F. Ramos, Universal bounds for the Littlewood-Paley first-order moments of the 3D Navier-Stokes equations, Comm. Math. Phys., 300 (2010), 301-315.  doi: 10.1007/s00220-010-1098-4.  Google Scholar

[24]

S. R. Spangler and C. R. Gwinn, Evidence for an inner scale to the density turbulence in the interstellar medium, Astrophys. J., 353 (1990), L29–L32. doi: 10.1086/185700.  Google Scholar

[25]

J. Wu, Regularity criteria for the generalized MHD equations, Comm. Partial Differential Equations, 33 (2008), 285-306.  doi: 10.1080/03605300701382530.  Google Scholar

show all references

References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, 343, Springer-Verlag, 2011. doi: 10.1007%2F978-3-642-16830-7.  Google Scholar

[2]

A. Basu and J. K. Bhattacharjee, Universal properties of three-dimensional magnetohydrodynamic turbulence: do Alfvén waves matter?, J. Stat. Mech., 2005 (2005), P07002. doi: 10.1088/1742-5468/2005/07/P07002.  Google Scholar

[3]

A. Basu, A. Sain, S. K. Dhar and R. Pandit, Multiscaling in models of magnetohydrodynamic turbulence, Phys. Rev. Lett., 81 (1998), 2687-2690.  doi: 10.1103/PhysRevLett.81.2687.  Google Scholar

[4]

D. Biskamp and W-C. Müller, Scaling properties of three-dimensional isotropic magnetohydrodynamic turbulence, Phys. Plasmas, 7 (2000), 4889-4900.  doi: 10.1063/1.1322562.  Google Scholar

[5]

M. Cannone, Harmonic analysis tools for solving incompressible Navier-Stokes equations, Handbook of Mathmatical Fluid Dynamics vol 3,161–244, North-Holland, Amsterdam, 2004.  Google Scholar

[6]

Q. Chen, C. Miao and Z. Zhang, A new Bernstein's inequality and the 2D dissipative quasi-geostrophic equation, Comm. Math. Phys., 271 (2007), 821-838.  doi: 10.1007/s00220-007-0193-7.  Google Scholar

[7]

Q. Chen, C. Miao and Z. Zhang, On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations, Comm. Math. Phys., 284 (2008), 919-930.  doi: 10.1007/s00220-008-0545-y.  Google Scholar

[8]

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 (2010), 561-578.  doi: 10.1007/s00205-008-0213-6.  Google Scholar

[9]

J. Cho, E. T. Vishniac and A. Lazarian, Simulations of magnetohydrodynamic turbulence in a strongly magnetized medium, Astrophys. J., 564 (2002), 291-301.  doi: 10.1086/324186.  Google Scholar

[10]

J. Cho and E. T. Vishniac, The anisotropy of magnetohydrodynamic Alfvénic turbulence, Astrophys. J., 539 (2000), 273-282.  doi: 10.1086/309213.  Google Scholar

[11]

P. Constantin, Euler equations, Navier-Stokes equations and turbulence, in Mathematical Foundation of Turbulent Viscous Flows, Lecture Notes in Math. Vol. 1871, Berlin: Springer, 2006, 1–43. doi: 10.1007%2F11545989_1.  Google Scholar

[12]

P. Constantin, The Littlewood-Paley spectrum in two-dimensional turbulence, Theor. Comput. Fluid Dyn., 9 (1997), 183-189.  doi: 10.1007/s001620050039.  Google Scholar

[13]

E. Falgarone and T. Passot, Turbulence and Magnetic Fields in Astrophysics, Lecture Notes in Physics, Springer, 2003. doi: 10.1007%2F3-540-36238-X.  Google Scholar

[14]

Y. Gupta, B. J. Rickett and W. A. Coles, Refractive interstellar scintillation of pulsar intensities at 74 MHz, Astrophysical J., 403 (1993), 183-201.  doi: 10.1086/172193.  Google Scholar

[15]

E. Hopf, Über die anfangswertaufgabe für die hydrodynamischen grundgleichungen, Math. Nachr., 4 (1951), 213-231.  doi: 10.1002/mana.3210040121.  Google Scholar

[16]

P. S. Iroshnikov, Turbulence of a conducting fluid in a strong magnetic field, Soviet Astronom. AJ, 7 (1964), 566-571.   Google Scholar

[17]

A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Proceedings of the Royal Society A, 434 (1991), 9-13.  doi: 10.1098/rspa.1991.0075.  Google Scholar

[18]

A. N. Kolmogorov, Dissipation of energy in the locally isotropic turbulence, Proceedings of the Royal Society A, 434 (1991), 15-17.  doi: 10.1098/rspa.1991.0076.  Google Scholar

[19]

R. H. Kraichnan, Lagrangian-history closure approximation for turbulence, Phys. Fluids, 8 (1965), 575-598.  doi: 10.1063/1.1761271.  Google Scholar

[20]

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

[21] C. Miao, J. Wu and Z. Zhang, Littlewood-Paley Theory and Applications to Fluid Dynamics Equations, Monographs on Modern pure mathematics, No. 142, Beijing: Science Press, 2012.   Google Scholar
[22]

W-C. Müller and D. Biskamp, Scaling properties of three-dimensional magnetohydrodynamic turbulence, Phys. Rev. Lett., 84 (2000), 475-478.  doi: 10.1103/PhysRevLett.84.475.  Google Scholar

[23]

F. Otto and F. Ramos, Universal bounds for the Littlewood-Paley first-order moments of the 3D Navier-Stokes equations, Comm. Math. Phys., 300 (2010), 301-315.  doi: 10.1007/s00220-010-1098-4.  Google Scholar

[24]

S. R. Spangler and C. R. Gwinn, Evidence for an inner scale to the density turbulence in the interstellar medium, Astrophys. J., 353 (1990), L29–L32. doi: 10.1086/185700.  Google Scholar

[25]

J. Wu, Regularity criteria for the generalized MHD equations, Comm. Partial Differential Equations, 33 (2008), 285-306.  doi: 10.1080/03605300701382530.  Google Scholar

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