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Global boundedness of solutions to the two-dimensional forager-exploiter model with logistic source
The Littlewood-Paley $ pth $-order moments in three-dimensional MHD turbulence
School of Mathematics and Systems Science, Beihang University, Beijing 100191, China |
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.
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. 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. |
| [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. 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. |
| [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. |
| [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. |
| [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. |
| [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. 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. |
| [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. 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.
|
| [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. 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. |
| [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. 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. |
| [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. |
| [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. |
| [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. |
| [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. 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. |
| [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. 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.
|
| [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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