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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

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.

Citation: Yao Nie, Jia Yuan. The Littlewood-Paley $ pth $-order moments in three-dimensional MHD turbulence. Discrete and 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.

[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.

[3]

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

[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.

[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.

[6]

Q. ChenC. 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.

[7]

Q. ChenC. 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.

[8]

Q. ChenC. 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.

[9]

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

[10]

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

[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.

[12]

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

[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.

[14]

Y. GuptaB. 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.

[15]

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

[16]

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

[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.

[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.

[19]

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

[20]

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

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

[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.

[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.

[25]

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

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.

[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.

[3]

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

[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.

[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.

[6]

Q. ChenC. 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.

[7]

Q. ChenC. 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.

[8]

Q. ChenC. 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.

[9]

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

[10]

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

[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.

[12]

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

[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.

[14]

Y. GuptaB. 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.

[15]

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

[16]

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

[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.

[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.

[19]

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

[20]

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

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

[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.

[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.

[25]

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

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