Reduced models for electron magnetohydrodynamics: Well-posedness and singularity formation

Dai, Mimi

doi:10.1090/tran/9427

Reduced models for electron magnetohydrodynamics: Well-posedness and singularity formation
HTML articles powered by AMS MathViewer

by Mimi Dai;

Trans. Amer. Math. Soc. 378 (2025), 3981-4009

DOI: https://doi.org/10.1090/tran/9427

Published electronically: March 25, 2025

PDF | Request permission

Abstract:

We propose one-dimensional reduced models for the three-dimensional electron magnetohydrodynamics which involves a highly nonlinear Hall term with intricate structure. The models contain nonlocal nonlinear terms which are more singular than that of the one-dimensional models for the Euler equation and the surface quasi-geostrophic equation. Local well-posedness is obtained in certain circumstances. Moreover, for a model with nonlocal transport term, we show that singularity develops in finite time for a class of initial data.

References

Marion Acheritogaray, Pierre Degond, Amic Frouvelle, and Jian-Guo Liu, Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system, Kinet. Relat. Models 4 (2011), no. 4, 901–918. MR 2861579, DOI 10.3934/krm.2011.4.901
Gregory R. Baker, Xiao Li, and Anne C. Morlet, Analytic structure of two $1$D-transport equations with nonlocal fluxes, Phys. D 91 (1996), no. 4, 349–375. MR 1382265, DOI 10.1016/0167-2789(95)00271-5
A. Castro and D. Córdoba, Global existence, singularities and ill-posedness for a nonlocal flux, Adv. Math. 219 (2008), no. 6, 1916–1936. MR 2456270, DOI 10.1016/j.aim.2008.07.015
Dongho Chae, Antonio Córdoba, Diego Córdoba, and Marco A. Fontelos, Finite time singularities in a 1D model of the quasi-geostrophic equation, Adv. Math. 194 (2005), no. 1, 203–223. MR 2141858, DOI 10.1016/j.aim.2004.06.004
J. Chen, On the regularity of the De Gregorio model for the 3D Euler equations, arXiv:2107.04777, 2021.
Jiajie Chen, On the slightly perturbed De Gregorio model on $S^1$, Arch. Ration. Mech. Anal. 241 (2021), no. 3, 1843–1869. MR 4284536, DOI 10.1007/s00205-021-01685-w
Jiajie Chen, Singularity formation and global well-posedness for the generalized Constantin-Lax-Majda equation with dissipation, Nonlinearity 33 (2020), no. 5, 2502–2532. MR 4105366, DOI 10.1088/1361-6544/ab74b0
Jiajie Chen, Thomas Y. Hou, and De Huang, On the finite time blowup of the De Gregorio model for the 3D Euler equations, Comm. Pure Appl. Math. 74 (2021), no. 6, 1282–1350. MR 4242826, DOI 10.1002/cpa.21991
Kyudong Choi, Thomas Y. Hou, Alexander Kiselev, Guo Luo, Vladimir Sverak, and Yao Yao, On the finite-time blowup of a one-dimensional model for the three-dimensional axisymmetric Euler equations, Comm. Pure Appl. Math. 70 (2017), no. 11, 2218–2243. MR 3707493, DOI 10.1002/cpa.21697
Kyudong Choi, Alexander Kiselev, and Yao Yao, Finite time blow up for a 1D model of 2D Boussinesq system, Comm. Math. Phys. 334 (2015), no. 3, 1667–1679. MR 3312447, DOI 10.1007/s00220-014-2146-2
P. Constantin, P. D. Lax, and A. Majda, A simple one-dimensional model for the three-dimensional vorticity equation, Comm. Pure Appl. Math. 38 (1985), no. 6, 715–724. MR 812343, DOI 10.1002/cpa.3160380605
Peter Constantin and Jiahong Wu, Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation, Ann. Inst. H. Poincaré C Anal. Non Linéaire 25 (2008), no. 6, 1103–1110. MR 2466323, DOI 10.1016/j.anihpc.2007.10.001
Antonio Córdoba and Diego Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys. 249 (2004), no. 3, 511–528. MR 2084005, DOI 10.1007/s00220-004-1055-1
Antonio Córdoba, Diego Córdoba, and Marco A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math. (2) 162 (2005), no. 3, 1377–1389. MR 2179734, DOI 10.4007/annals.2005.162.1377
Antonio Córdoba, Diego Córdoba, and Marco A. Fontelos, Integral inequalities for the Hilbert transform applied to a nonlocal transport equation, J. Math. Pures Appl. (9) 86 (2006), no. 6, 529–540 (English, with English and French summaries). MR 2281451, DOI 10.1016/j.matpur.2006.08.002
Mimi Dai, Almost sure existence of global weak solutions for supercritical electron MHD, Discrete Contin. Dyn. Syst. Ser. B 29 (2024), no. 3, 1584–1610. MR 4696115, DOI 10.3934/dcdsb.2023146
Salvatore De Gregorio, A partial differential equation arising in a $1$D model for the $3$D vorticity equation, Math. Methods Appl. Sci. 19 (1996), no. 15, 1233–1255. MR 1410208, DOI 10.1002/(SICI)1099-1476(199610)19:15<1233::AID-MMA828>3.3.CO;2-N
Salvatore De Gregorio, On a one-dimensional model for the three-dimensional vorticity equation, J. Statist. Phys. 59 (1990), no. 5-6, 1251–1263. MR 1063199, DOI 10.1007/BF01334750
Tam Do, On a 1D transport equation with nonlocal velocity and supercritical dissipation, J. Differential Equations 256 (2014), no. 9, 3166–3178. MR 3171771, DOI 10.1016/j.jde.2014.01.037
Hongjie Dong, Well-posedness for a transport equation with nonlocal velocity, J. Funct. Anal. 255 (2008), no. 11, 3070–3097. MR 2464570, DOI 10.1016/j.jfa.2008.08.005
Hongjie Dong and Dong Li, On a one-dimensional $\alpha$-patch model with nonlocal drift and fractional dissipation, Trans. Amer. Math. Soc. 366 (2014), no. 4, 2041–2061. MR 3152722, DOI 10.1090/S0002-9947-2013-06075-8
Tarek M. Elgindi, Tej-Eddine Ghoul, and Nader Masmoudi, Stable self-similar blow-up for a family of nonlocal transport equations, Anal. PDE 14 (2021), no. 3, 891–908. MR 4259877, DOI 10.2140/apde.2021.14.891
Tarek M. Elgindi and In-Jee Jeong, On the effects of advection and vortex stretching, Arch. Ration. Mech. Anal. 235 (2020), no. 3, 1763–1817. MR 4065651, DOI 10.1007/s00205-019-01455-9
Jan S. Hesthaven, Sigal Gottlieb, and David Gottlieb, Spectral methods for time-dependent problems, Cambridge Monographs on Applied and Computational Mathematics, vol. 21, Cambridge University Press, Cambridge, 2007. MR 2333926, DOI 10.1017/CBO9780511618352
Thomas Y. Hou, Congming Li, Zuoqiang Shi, Shu Wang, and Xinwei Yu, On singularity formation of a nonlinear nonlocal system, Arch. Ration. Mech. Anal. 199 (2011), no. 1, 117–144. MR 2754339, DOI 10.1007/s00205-010-0319-5
Thomas Y. Hou and Zhen Lei, On the stabilizing effect of convection in three-dimensional incompressible flows, Comm. Pure Appl. Math. 62 (2009), no. 4, 501–564. MR 2492706, DOI 10.1002/cpa.20254
In-Jee Jeong and Sung-Jin Oh, On the Cauchy problem for the Hall and electron magnetohydrodynamic equations without resistivity I: Illposedness near degenerate stationary solutions, Ann. PDE 8 (2022), no. 2, Paper No. 15, 106. MR 4456288, DOI 10.1007/s40818-022-00134-5
H. Jia, S. Stewart, and V. Sverak, On the De Gregorio modification of the Constantin-Lax-Majda model, Arch. Ration. Mech. Anal. 231 (2019), no. 2, 1269–1304. MR 3900823, DOI 10.1007/s00205-018-1298-1
Tosio Kato and Chi Yuen Lai, Nonlinear evolution equations and the Euler flow, J. Funct. Anal. 56 (1984), no. 1, 15–28. MR 735703, DOI 10.1016/0022-1236(84)90024-7
Zhen Lei, Jie Liu, and Xiao Ren, On the Constantin-Lax-Majda model with convection, Comm. Math. Phys. 375 (2020), no. 1, 765–783. MR 4082178, DOI 10.1007/s00220-019-03584-4
Dong Li and Jose Rodrigo, Blow-up of solutions for a 1D transport equation with nonlocal velocity and supercritical dissipation, Adv. Math. 217 (2008), no. 6, 2563–2568. MR 2397459, DOI 10.1016/j.aim.2007.11.002
Dong Li and José L. Rodrigo, On a one-dimensional nonlocal flux with fractional dissipation, SIAM J. Math. Anal. 43 (2011), no. 1, 507–526. MR 2783214, DOI 10.1137/100794924
G. Luo and T. Y. Hou, Potentially singular solutions of the 3D incompressible Euler equations, PNAS 111 (2014), no. 36, 12968–12973.
Hideyuki Miura, Dissipative quasi-geostrophic equation for large initial data in the critical Sobolev space, Comm. Math. Phys. 267 (2006), no. 1, 141–157. MR 2238907, DOI 10.1007/s00220-006-0023-3
Anne C. Morlet, Further properties of a continuum of model equations with globally defined flux, J. Math. Anal. Appl. 221 (1998), no. 1, 132–160. MR 1619138, DOI 10.1006/jmaa.1997.5801
Hisashi Okamoto, Takashi Sakajo, and Marcus Wunsch, On a generalization of the Constantin-Lax-Majda equation, Nonlinearity 21 (2008), no. 10, 2447–2461. MR 2439488, DOI 10.1088/0951-7715/21/10/013
Thomas Runst and Winfried Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, De Gruyter Series in Nonlinear Analysis and Applications, vol. 3, Walter de Gruyter & Co., Berlin, 1996. MR 1419319, DOI 10.1515/9783110812411
Takashi Sakajo, On global solutions for the Constantin-Lax-Majda equation with a generalized viscosity term, Nonlinearity 16 (2003), no. 4, 1319–1328. MR 1986297, DOI 10.1088/0951-7715/16/4/307
Luis Silvestre and Vlad Vicol, On a transport equation with nonlocal drift, Trans. Amer. Math. Soc. 368 (2016), no. 9, 6159–6188. MR 3461030, DOI 10.1090/tran6651
A. Zygmund, Trigonometric Series, Cambridge University Press. Third Edition, 2002.

AMS Website Down

Transactions of the American Mathematical Society

Reduced models for electron magnetohydrodynamics: Well-posedness and singularity formation
HTML articles powered by AMS MathViewer

Abstract: