Abstract
We study two dyadic models for incompressible ideal magnetohydrodynamics, one with a uni-directional energy cascade and the other one with both forward and backward energy cascades. Global existence of weak solutions and local well-posedness are established for both models. In addition, solutions to the model with uni-directional energy cascade associated with positive initial data are shown to develop blow-up at a finite time. Moreover, a set of fixed points is found for each model. Linear instability about some particular fixed points is proved.
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Alexakis, A., Mininni, P.D., Pouquet, A.: Shell-to-shell energy transfer in magnetohydrodynamics. I. Steady state turbulence. Phys. Rev. E 72(4), 046301 (2005)
Antonov, T., Frick, P.: Cascade and scaling in a class of shell models for MHD-turbulence. Math. Model. Syst. Precesses 8, 4–10 (2000)
Antonov, T., Lozhkin, S., Frick, P., Sokoloff, D.: A shell model for free decaying MHD-turbulence and the role of the magnetic Prandtl number. Magnetohydrodynamics 37, 87–92 (2001)
Barbato, D., Flandoli, F., Morandin, F.: A theorem of uniqueness for an inviscid dyadic model. C.R. Math. Acad. Sci. Paris 348(9–10), 525–528 (2010)
Barbato, D., Flandoli, F., Morandin, F.: Anomalous dissipation in a stochastic inviscid dyadic model. Ann. Appl. Probab. 21(6), 2424–2446 (2011)
Barbato, D., Flandoli, F., Morandin, F.: Energy dissipation and self-similar solutions for an unforced inviscid dyadic model. Trans. Am. Math. Soc. 363(4), 1925–1946 (2011)
Barbato, D., Flandoli, F., Morandin, F.: Uniqueness for a stochastic inviscid dyadic model. Proc. Am. Math. Soc. 138(7), 2607–2617 (2010)
Barbato, D., Morandin, F.: Positive and non-positive solutions for an inviscid dyadic model: well-posedness and regularity. Nonlinear Differ. Equ. Appl. 20(3), 1105–1123 (2013)
Basu, A., Sain, A., Dhar, S.K., Pandit, R.: Multiscaling in models of magnetohydrodynamic turbulence. Phys. Rev. Lett. 81, 2687–2690 (1998)
Biferale, L.: Shell models of energy cascade in turbulence. Annu. Rev. Fluid Mech. 35, 441468 (2003)
Biskamp, D.: Cascade models for magnetohydrodynamic turbulence. Phys. Rev. E 50, 2702–2711 (1994)
Caflisch, R.E., Klapper, I., Steele, G.: Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD. Commun. Math. Phys. 184, 443–455 (1997)
Cheskidov, A., Dai, M.: Kolmogorov’s dissipation number and the number of degrees of freedom for the 3D Navier–Stokes equations. Proc. R. Soc. Edinb. Sect. A 149(2), 429–446 (2019)
Cheskidov, A., Friedlander, S., Pavlović, N.: Inviscid dyadic model of turbulence: the fixed point and Onsager’s conjecture. J. Math. Phys. 48(6), 065503 (2007)
Cheskidov, A., Friedlander, S., Pavlović, N.: An inviscid dyadic model of turbulence: the global attractor. Discret. Contin. Dyn. Syst. 26(3), 781–794 (2010)
Cheskidov, A., Glatt-Holtz, N., Pavlović, N., Shvydkoy, R., Vicol, V.: Susan Friedlander’s contributions in mathematical fluid dynamics. Not. Am. Math. Soc. 68(3), 331–343 (2021)
Cheskidov, A., Zaya, K.: Regularizing effect of the forward energy cascade in the inviscid dyadic model. Proc. Am. Math. Soc. 144, 73–85 (2016)
Constantin, P., Levant, B., Titi, E.: Analytic study of the shell model of turbulence. Physica D 219(2), 120–141 (2006)
Dai, M.: Blow-up of a dyadic model with intermittency dependence for the Hall MHD. Phys. D Nonlinear Phenom. (2021) (to appear)
Dai, M.: Blow-up of dyadic MHD models with forward energy cascade. ar**v: 2102.03498 (2021)
Dai, M., Friedlander, S.: Uniqueness and non-uniqueness results for dyadic MHD models. ar**v:2107.04073 (2021)
Desnyanskiy, V.N., Novikov, E.A.: Evolution of turbulence spectra toward a similarity regime. Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 10, 127–136 (1974)
Dinaburg, E.I., Sinai, Y.G.: A quasi-linear approximation of three-dimensional Navier–Stokes system. Moscow Math. J. 1, 381–388 (2001)
Frick, P., Stepanov, R., Sokoloff, D.: Large- and small-scale interactions and quenching in an \(\alpha ^2\)-dynamo. Phys. Rev. E 74(6), 066310 (2006)
Frik, P.G.: Two-dimensional MHD turbulence: a hierarchical model. Magnetohydrodynamics 20(3), 262–267 (1984)
Friedlander, S., Glatt-Holtz, N., Vicol, V.: Inviscid limits for a stochastically forced shell model of turbulent flow. Ann. Inst. Henri Poincaré Probab. Stat. 52(3), 1217–1247 (2016)
Friedlander, S., Pavlović, N.: Blowup in a three-dimensional vector model for the Euler equations. Commun. Pure Appl. Math. 57(6), 705–725 (2004)
Gledzer, E.B.: System of hydrodynamic type admitting two quadratic integrals of motion. Sov. Phys. Dokl. 18, 216–217 (1973)
Gloaguen, C., Léorat, J., Pouquet, A., Grappin, R.: A scalar model for MHD turbulence. Physica D 17(2), 154–182 (1985)
Jeong, I., Li, D.: A blow-up result for dyadic models of the Euler equations. Commun. Math. Phys. 337, 1027–1034 (2015)
Kang, E., Lee, J.: Remarks on the magnetic helicity and energy conservation for ideal magneto-hydrodynamics. Nonlinearity 20(11), 2681–2689 (2007)
Katz, N., Pavlović, N.: Finite time blow-up for a dyadic model of the Euler equations. Trans. Am. Math. Soc. 357(2), 695–708 (2005)
Kiselev, A., Zlatoš, A.: On discrete models of the Euler equation. Int. Math. Res. Not. 38, 2315–2339 (2005)
Nigro, G., Malara, F., Carbone, V., Veltri, P.: Nanoflares and MHD turbulence in coronal loops: a hybrid shell model. Phys. Rev. Lett. 92(19), 194501 (2004)
Obukhov, A.M.: Some general properties of equations describing the dynamics of the atmosphere. Izv. Akad. Nauk. SSSR Ser. Fiz. Atmos. Okeana 7, 695–704 (1971)
Ohkitani, K., Yamada, M.: Temporal intermittency in the energy cascade process and local Lyapunov analysis in fully-developed model of turbulence. Prog. Theor. Phys. 81, 329–341 (1989)
Plunian, F., Stepanov, R., Frick, P.: Shell models of magnetohydrodynamic turbulence. Phys. Rep. 523, 1–60 (2013)
Romito, M.: Uniqueness and blow-up for a stochastic viscous dyadic model. Probab. Theor. Relat. Fields 158, 895–924 (2014)
Acknowledgements
The work of M. Dai is partially supported by NSF Grants DMS-1815069 and DMS-2009422; the work of S. Friedlander is partially supported by NSF Grant DMS-1613135. S. Friedlander is grateful to IAS for its hospitality in 2020-2021.
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Dai, M., Friedlander, S. Dyadic models for ideal MHD. J. Math. Fluid Mech. 24, 21 (2022). https://doi.org/10.1007/s00021-021-00640-9
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DOI: https://doi.org/10.1007/s00021-021-00640-9