{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T19:13:49Z","timestamp":1775848429980,"version":"3.50.1"},"reference-count":32,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,3,9]],"date-time":"2025-03-09T00:00:00Z","timestamp":1741478400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,3,9]],"date-time":"2025-03-09T00:00:00Z","timestamp":1741478400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001659","name":"German Research Foundation","doi-asserted-by":"crossref","award":["258734477"],"award-info":[{"award-number":["258734477"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001659","name":"German Research Foundation","doi-asserted-by":"crossref","award":["258734477"],"award-info":[{"award-number":["258734477"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2025,6]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>In this article, we consider the stability threshold of the 2D magnetohydrodynamics (MHD) equations near a combination of Couette flow and large constant magnetic field. We study the partial dissipation regime with full viscous and only horizontal magnetic dissipation. In particular, we show that this regime behaves qualitatively differently than both the fully dissipative and the non-resistive setting.<\/jats:p>","DOI":"10.1007\/s00332-025-10148-5","type":"journal-article","created":{"date-parts":[[2025,3,9]],"date-time":"2025-03-09T05:20:46Z","timestamp":1741497646000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["On the Sobolev Stability Threshold for the 2D MHD Equations with Horizontal Magnetic Dissipation"],"prefix":"10.1007","volume":"35","author":[{"given":"Niklas","family":"Knobel","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christian","family":"Zillinger","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,3,9]]},"reference":[{"issue":"4","key":"10148_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00030-022-00773-4","volume":"29","author":"D Adhikari","year":"2022","unstructured":"Adhikari, D., BenSaid, O., Pandey, U.R., Wu, J.: Stability and large-time behavior for the 2D Boussineq system with horizontal dissipation and vertical thermal diffusion. Nonlinear Differ. Equ. Appl. NoDEA 29(4), 1\u201343 (2022)","journal-title":"Nonlinear Differ. Equ. Appl. NoDEA"},{"issue":"1","key":"10148_CR2","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1090\/S0002-9947-1988-0920153-5","volume":"305","author":"C Bardos","year":"1988","unstructured":"Bardos, C., Sulem, C., Sulem, P.-L.: Longtime dynamics of a conductive fluid in the presence of a strong magnetic field. Trans. Am. Math. Soc. 305(1), 175\u2013191 (1988)","journal-title":"Trans. Am. Math. Soc."},{"key":"10148_CR3","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1007\/s10240-015-0070-4","volume":"122","author":"J Bedrossian","year":"2015","unstructured":"Bedrossian, J., Masmoudi, N.: Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations. Publ. Math. Inst. Hautes \u00c9tudes Sci. 122, 195\u2013300 (2015)","journal-title":"Publ. Math. Inst. Hautes \u00c9tudes Sci."},{"key":"10148_CR4","doi-asserted-by":"publisher","first-page":"541","DOI":"10.4007\/annals.2017.185.2.4","volume":"185","author":"J Bedrossian","year":"2017","unstructured":"Bedrossian, J., Germain, P., Masmoudi, N.: On the stability threshold for the 3D Couette flow in Sobolev regularity. Ann. Math. 185, 541\u2013608 (2017)","journal-title":"Ann. Math."},{"issue":"6","key":"10148_CR5","doi-asserted-by":"publisher","first-page":"2051","DOI":"10.1007\/s00332-016-9330-9","volume":"28","author":"J Bedrossian","year":"2018","unstructured":"Bedrossian, J., Vicol, V., Wang, F.: The Sobolev stability threshold for 2D shear flows near Couette. J. Nonlinear Sci. 28(6), 2051\u20132075 (2018)","journal-title":"J. Nonlinear Sci."},{"issue":"12","key":"10148_CR6","doi-asserted-by":"publisher","first-page":"3685","DOI":"10.1002\/cpa.22123","volume":"76","author":"J Bedrossian","year":"2023","unstructured":"Bedrossian, J., Bianchini, R., Zelati, M.C., Dolce, M.: Nonlinear inviscid damping and shear-buoyancy instability in the two-dimensional Boussinesq equations. Commun. Pure Appl. Math. 76(12), 3685\u20133768 (2023)","journal-title":"Commun. Pure Appl. Math."},{"key":"10148_CR7","unstructured":"Bianchini, R., Zelati, M.C., Dolce, M.: Linear inviscid damping for shear flows near Couette in the 2D stably stratified regime. arXiv preprint https:\/\/doi.org\/10.48550\/arXiv.2005.09058 (2020)"},{"issue":"3","key":"10148_CR8","doi-asserted-by":"publisher","first-page":"985","DOI":"10.1007\/s00205-013-0610-3","volume":"208","author":"C Cao","year":"2013","unstructured":"Cao, C., Jiahong, W.: Global regularity for the two-dimensional anisotropic Boussinesq equations with vertical dissipation. Arch. Ration. Mech. Anal. 208(3), 985\u20131004 (2013)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"7","key":"10148_CR9","doi-asserted-by":"publisher","first-page":"2661","DOI":"10.1016\/j.jde.2013.01.002","volume":"254","author":"C Cao","year":"2013","unstructured":"Cao, C., Regmi, D., Jiahong, W.: The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion. J. Differ. Equ. 254(7), 2661\u20132681 (2013)","journal-title":"J. Differ. Equ."},{"issue":"4","key":"10148_CR10","doi-asserted-by":"publisher","first-page":"863","DOI":"10.1063\/1.859841","volume":"3","author":"XL Chen","year":"1991","unstructured":"Chen, X.L., Morrison, P.J.: A sufficient condition for the ideal instability of shear flow with parallel magnetic field. Phys. Fluids B Plasma Phys. 3(4), 863\u2013865 (1991)","journal-title":"Phys. Fluids B Plasma Phys."},{"key":"10148_CR11","doi-asserted-by":"publisher","DOI":"10.1017\/9781316672853","volume-title":"Introduction to Magnetohydrodynamics. Cambridge Texts in Applied Mathematics","author":"PA Davidson","year":"2016","unstructured":"Davidson, P.A.: Introduction to Magnetohydrodynamics. Cambridge Texts in Applied Mathematics, 2nd edn. Cambridge University Press (2016)","edition":"2"},{"key":"10148_CR12","unstructured":"Deng, Y., Masmoudi, N.: Long-time instability of the Couette flow in low Gevrey spaces. Commun. Pure Appl. Math. (2018)"},{"issue":"1","key":"10148_CR13","doi-asserted-by":"publisher","first-page":"643","DOI":"10.1007\/s00205-021-01697-6","volume":"242","author":"Yu Deng","year":"2021","unstructured":"Deng, Yu., Zillinger, C.: Echo chains as a linear mechanism: norm inflation, modified exponents and asymptotics. Arch. Ration. Mech. Anal. 242(1), 643\u2013700 (2021)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"12","key":"10148_CR14","doi-asserted-by":"publisher","first-page":"109255","DOI":"10.1016\/j.jfa.2021.109255","volume":"281","author":"W Deng","year":"2021","unstructured":"Deng, W., Jiahong, W., Zhang, P.: Stability of Couette flow for 2D Boussinesq system with vertical dissipation. J. Funct. Anal. 281(12), 109255 (2021)","journal-title":"J. Funct. Anal."},{"key":"10148_CR15","doi-asserted-by":"crossref","unstructured":"Dolce, M.: Stability threshold of the 2D Couette flow in a homogeneous magnetic field using symmetric variables. arxiv preprint https:\/\/doi.org\/10.48550\/arXiv.2308.12589 (2023)","DOI":"10.1007\/s00220-024-04982-z"},{"issue":"1","key":"10148_CR16","doi-asserted-by":"publisher","first-page":"012107","DOI":"10.1063\/1.1834591","volume":"12","author":"M Hirota","year":"2005","unstructured":"Hirota, M., Tatsuno, T., Yoshida, Z.: Resonance between continuous spectra: secular behavior of Alfv\u00e9n waves in a flowing plasma. Phys. Plasmas 12(1), 012107 (2005)","journal-title":"Phys. Plasmas"},{"issue":"2010","key":"10148_CR17","doi-asserted-by":"publisher","first-page":"1365","DOI":"10.1098\/rspa.2000.0725","volume":"457","author":"DW Hughes","year":"2001","unstructured":"Hughes, D.W., Tobias, S.M.: On the instability of magnetohydrodynamic shear flows. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 457(2010), 1365\u20131384 (2001)","journal-title":"Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci."},{"issue":"10","key":"10148_CR18","doi-asserted-by":"publisher","first-page":"105209","DOI":"10.1063\/1.5051624","volume":"8","author":"Z Hussain","year":"2018","unstructured":"Hussain, Z., Hussain, S., Kong, T., Liu, Z.: Instability of MHD Couette flow of an electrically conducting fluid. AIP Adv. 8(10), 105209 (2018)","journal-title":"AIP Adv."},{"key":"10148_CR19","doi-asserted-by":"publisher","first-page":"244","DOI":"10.1016\/j.aml.2019.03.013","volume":"94","author":"R Ji","year":"2019","unstructured":"Ji, R., Lin, H., Jiahong, W., Yan, L.: Stability for a system of the 2D magnetohydrodynamic equations with partial dissipation. Appl. Math. Lett. 94, 244\u2013249 (2019)","journal-title":"Appl. Math. Lett."},{"key":"10148_CR20","doi-asserted-by":"publisher","first-page":"625","DOI":"10.1016\/j.jde.2023.05.020","volume":"367","author":"N Knobel","year":"2023","unstructured":"Knobel, N., Zillinger, C.: On echoes in magnetohydrodynamics with magnetic dissipation. J. Differ. Equ. 367, 625\u2013688 (2023)","journal-title":"J. Differ. Equ."},{"issue":"1","key":"10148_CR21","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00332-020-09672-3","volume":"31","author":"S Lai","year":"2021","unstructured":"Lai, S., Jiahong, W., Xiaojing, X., Zhang, J., Zhong, Y.: Optimal decay estimates for 2D Boussinesq equations with partial dissipation. J. Nonlinear Sci. 31(1), 1\u201333 (2021)","journal-title":"J. Nonlinear Sci."},{"key":"10148_CR22","doi-asserted-by":"publisher","first-page":"859","DOI":"10.1007\/s00220-020-03768-3","volume":"377","author":"K Liss","year":"2020","unstructured":"Liss, K.: On the Sobolev stability threshold of 3D Couette flow in a uniform magnetic field. Commun. Math. Phys. 377, 859\u2013908 (2020)","journal-title":"Commun. Math. Phys."},{"key":"10148_CR23","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511613203","volume-title":"Vorticity and Incompressible Flow","author":"AJ Majda","year":"2001","unstructured":"Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow, vol. 27. Cambridge University Press (2001)"},{"issue":"2","key":"10148_CR24","doi-asserted-by":"publisher","first-page":"245","DOI":"10.4171\/aihpc\/8","volume":"39","author":"N Masmoudi","year":"2022","unstructured":"Masmoudi, N., Zhao, W.: Stability threshold of two-dimensional Couette flow in Sobolev spaces. Ann. Inst. H. Poincar\u00e9 C Anal. Non Lin\u00e9aire 39(2), 245\u2013325 (2022)","journal-title":"Ann. Inst. H. Poincar\u00e9 C Anal. Non Lin\u00e9aire"},{"issue":"1","key":"10148_CR25","doi-asserted-by":"publisher","first-page":"109736","DOI":"10.1016\/j.jfa.2022.109736","volume":"284","author":"N Masmoudi","year":"2023","unstructured":"Masmoudi, N., Zhai, C., Zhao, W.: Asymptotic stability for two-dimensional Boussinesq systems around the Couette flow in a finite channel. J. Funct. Anal. 284(1), 109736 (2023)","journal-title":"J. Funct. Anal."},{"issue":"2","key":"10148_CR26","doi-asserted-by":"publisher","first-page":"503","DOI":"10.1016\/j.jfa.2014.04.020","volume":"267","author":"X Ren","year":"2014","unstructured":"Ren, X., Jiahong, W., Xiang, Z., Zhang, Z.: Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion. J. Funct. Anal. 267(2), 503\u2013541 (2014)","journal-title":"J. Funct. Anal."},{"issue":"2","key":"10148_CR27","doi-asserted-by":"publisher","first-page":"585","DOI":"10.1007\/s00205-020-01515-5","volume":"237","author":"L Tao","year":"2020","unstructured":"Tao, L., Jiahong, W., Zhao, K., Zheng, X.: Stability near hydrostatic equilibrium to the 2D Boussinesq equations without thermal diffusion. Arch. Ration. Mech. Anal. 237(2), 585\u2013630 (2020)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"2","key":"10148_CR28","doi-asserted-by":"publisher","first-page":"1284","DOI":"10.1137\/22M1495160","volume":"55","author":"C Zhai","year":"2023","unstructured":"Zhai, C., Zhao, W.: Stability threshold of the Couette flow for Navier\u2013Stokes Boussinesq system with large Richardson number $$\\gamma > 1\/4$$. SIAM J. Math. Anal. 55(2), 1284\u20131318 (2023)","journal-title":"SIAM J. Math. Anal."},{"key":"10148_CR29","doi-asserted-by":"publisher","first-page":"1317","DOI":"10.1007\/s00205-021-01706-8","volume":"242","author":"C Zhai","year":"2021","unstructured":"Zhai, C., Zhang, Z., Zhao, W.: Long-time behavior of Alfv\u00e9n waves in a flowing plasma: generation of the magnetic island. Arch. Ration. Mech. Anal. 242, 1317\u20131394 (2021)","journal-title":"Arch. Ration. Mech. Anal."},{"key":"10148_CR30","doi-asserted-by":"crossref","unstructured":"Zhao, W., Zi, R.: Asymptotic stability of Couette flow in a strong uniform magnetic field for the Euler-MHD system. arXiv preprint https:\/\/doi.org\/10.48550\/arXiv.2305.04052 (2023)","DOI":"10.1007\/s00205-024-01996-8"},{"key":"10148_CR31","unstructured":"Zillinger, C.: On echo chains in the linearized Boussinesq equations around traveling waves. arXiv preprint https:\/\/doi.org\/10.48550\/arXiv.2103.15441 (2021)"},{"issue":"4","key":"10148_CR32","doi-asserted-by":"publisher","first-page":"64","DOI":"10.1007\/s00332-021-09723-3","volume":"31","author":"C Zillinger","year":"2021","unstructured":"Zillinger, C.: On the Boussinesq equations with non-monotone temperature profiles. J. Nonlinear Sci. 31(4), 64 (2021)","journal-title":"J. Nonlinear Sci."}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-025-10148-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00332-025-10148-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-025-10148-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,7]],"date-time":"2025-05-07T16:08:18Z","timestamp":1746634098000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00332-025-10148-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,9]]},"references-count":32,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,6]]}},"alternative-id":["10148"],"URL":"https:\/\/doi.org\/10.1007\/s00332-025-10148-5","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"value":"0938-8974","type":"print"},{"value":"1432-1467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,3,9]]},"assertion":[{"value":"12 September 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 February 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 March 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"54"}}