{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T11:51:50Z","timestamp":1791546710526,"version":"4.3.4"},"reference-count":55,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T00:00:00Z","timestamp":1791504000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T00:00:00Z","timestamp":1791504000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/X020886\/1"],"award-info":[{"award-number":["EP\/X020886\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000761","name":"Imperial College London","doi-asserted-by":"publisher","award":["President\u2019s PhD Scholarships"],"award-info":[{"award-number":["President\u2019s PhD Scholarships"]}],"id":[{"id":"10.13039\/501100000761","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Ann. PDE"],"published-print":{"date-parts":[[2026,12]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We consider the kinematic dynamo equations for a passive vector in\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathcal {M} \\times \\mathbb {T} \\subseteq \\mathbb {R}^2 \\times \\mathbb {T}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>M<\/mml:mi>\n                            <mml:mo>\u00d7<\/mml:mo>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mo>\u2286<\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow>\n                                <mml:mi>R<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>\u00d7<\/mml:mo>\n                            <mml:mi>T<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    describing the evolution of a magnetic field with resistivity\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varepsilon &gt; 0$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>\u03b5<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , that is transported by a given velocity field. For a broad class of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$C^3$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msup>\n                            <mml:mi>C<\/mml:mi>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:msup>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    velocity fields with helical geometry, we establish the existence of solutions that exhibit exponential growth over time. We construct an unstable eigenmode via detailed resolvent estimates of the corresponding linear operator, which we carry out by introducing suitable Green\u2019s functions that accurately approximate the local behaviour of the true system. This approach yields an explicit asymptotic expression for the growing mode, providing a sharp description of the instability mechanism. We first derive the results with\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathcal {M}=\\mathbb {R}^2$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>M<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow>\n                                <mml:mi>R<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    for a large class of velocity fields that includes finite energy examples. We then consider the case of domains with boundary, where\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathcal {M}\\times \\mathbb {T}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>M<\/mml:mi>\n                            <mml:mo>\u00d7<\/mml:mo>\n                            <mml:mi>T<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    denotes a periodic cylinder, annular cylinder, or the exterior of a cylinder, with the boundary conditions of perfectly conducting walls. Our results offer a rigorous and sharp mathematical justification for the physically conjectured process by which helical flows can sustain magnetic field generation in the Ponomarenko dynamo, with growth rate of order\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varepsilon ^{1\/3}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msup>\n                            <mml:mi>\u03b5<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo>\/<\/mml:mo>\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    .\n                  <\/jats:p>","DOI":"10.1007\/s40818-026-00255-1","type":"journal-article","created":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T10:09:43Z","timestamp":1791540583000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Spectral Instability in the Smooth Ponomarenko Dynamo"],"prefix":"10.1007","volume":"12","author":[{"given":"V\u00edctor","family":"Navarro-Fern\u00e1ndez","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"David","family":"Villringer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,10,9]]},"reference":[{"issue":"3","key":"255_CR1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jfa.2022.109522","volume":"283","author":"D Albritton","year":"2022","unstructured":"Albritton, D., Beekie, R., Novack, M.: Enhanced dissipation and H\u00f6rmander\u2019s hypoellipticity. J. Funct. Anal. 283(3), 109522 (2022)","journal-title":"J. Funct. Anal."},{"key":"255_CR2","volume-title":"Vector-Valued Laplace Transforms and Cauchy Problems","author":"W Arendt","year":"2013","unstructured":"Arendt, W., Batty, C.J.K., Neubrander, F.: Vector-Valued Laplace Transforms and Cauchy Problems. Springer, Berlin (2013)"},{"key":"255_CR3","volume-title":"Arnold\u2019s Problems","author":"VI Arnold","year":"2004","unstructured":"Arnold, V.I.: Arnold\u2019s Problems. Springer, Berlin (2004)"},{"key":"255_CR4","series-title":"Applied Mathematical Sciences","doi-asserted-by":"crossref","DOI":"10.1007\/b97593","volume-title":"Topological Methods in Hydrodynamics","author":"VI Arnold","year":"1998","unstructured":"Arnold, V.I., Khesin, B.A.: Topological Methods in Hydrodynamics. Applied Mathematical Sciences, vol. 125. Springer, New York (1998)"},{"issue":"1\u20134","key":"255_CR5","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1080\/03091929308203618","volume":"73","author":"PH Baxendale","year":"1993","unstructured":"Baxendale, P.H., Rozovskii, B.L.: Kinematic dynamo and intermittence in a turbulent flow. Geophys. Astrophys. Fluid Dyn. 73(1\u20134), 33\u201360 (1993)","journal-title":"Geophys. Astrophys. Fluid Dyn."},{"issue":"3","key":"255_CR6","doi-asserted-by":"publisher","first-page":"1161","DOI":"10.1007\/s00205-017-1099-y","volume":"224","author":"J Bedrossian","year":"2017","unstructured":"Bedrossian, J., Coti Zelati, M.: Enhanced dissipation, hypoellipticity, and anomalous small noise inviscid limits in shear flows. Arch. Ration. Mech. Anal. 224(3), 1161\u20131204 (2017)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"1\u20134","key":"255_CR7","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.physrep.2005.06.005","volume":"417","author":"A Brandenburg","year":"2005","unstructured":"Brandenburg, A., Subramanian, K.: Astrophysical magnetic fields and nonlinear dynamo theory. Phys. Rep. 417(1\u20134), 1\u2013209 (2005)","journal-title":"Phys. Rep."},{"issue":"2","key":"255_CR8","doi-asserted-by":"publisher","first-page":"172","DOI":"10.1016\/0031-9201(79)90039-6","volume":"20","author":"S Childress","year":"1979","unstructured":"Childress, S.: Alpha-effect in flux ropes and sheets. Phys. Earth Planet. Int. 20(2), 172\u2013180 (1979)","journal-title":"Phys. Earth Planet. Int."},{"key":"255_CR9","volume-title":"Stretch, Twist, Fold: The Fast Dynamo, First, Monographs","author":"S Childress","year":"2008","unstructured":"Childress, S., Gilbert, A.D.: Stretch, Twist, Fold: The Fast Dynamo, First, Monographs, vol. 37. Springer, Berlin (2008)"},{"key":"255_CR10","unstructured":"Coti Zelati, M., Navarro-Fern\u00e1ndez, V.: Three-dimensional exponential mixing and ideal kinematic dynamo with randomized ABC flows. arXiv:2407.18028 (2024)"},{"key":"255_CR11","unstructured":"Coti Zelati, M., Sorella, M., Villringer, D.: Alpha-unstable flows and the fast dynamo problem. arXiv:2504.00855 (2025)"},{"key":"255_CR12","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1093\/mnras\/94.1.39","volume":"94","author":"TG Cowling","year":"1933","unstructured":"Cowling, T.G.: The magnetic field of sunspots. Mon. Not. R. Astron. Soc. 94, 39\u201348 (1933)","journal-title":"Mon. Not. R. Astron. Soc."},{"issue":"1","key":"255_CR13","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1016\/s0294-1449(02)00009-4","volume":"20","author":"B Desjardins","year":"2003","unstructured":"Desjardins, B., Grenier, E.: Linear instability implies nonlinear instability for various types of viscous boundary layers. Annales de l\u2019Institut Henri Poincar\u00e9 C, Analyse non lin\u00e9aire 20(1), 87\u2013106 (2003)","journal-title":"Annales de l\u2019Institut Henri Poincar\u00e9 C, Analyse non lin\u00e9aire"},{"key":"255_CR14","series-title":"Student\u2019s Guides","doi-asserted-by":"publisher","DOI":"10.1017\/9781009089531","volume-title":"A Student\u2019s Guide to Laplace Transforms","author":"D Fleisch","year":"2022","unstructured":"Fleisch, D.: A Student\u2019s Guide to Laplace Transforms. Student\u2019s Guides, Cambridge University Press (2022)"},{"key":"255_CR15","volume-title":"Introduction to Partial Differential Equations","author":"GB Folland","year":"1996","unstructured":"Folland, G.B.: Introduction to Partial Differential Equations, vol. 2. Princeton University Press (1996)"},{"issue":"2","key":"255_CR16","doi-asserted-by":"publisher","first-page":"335","DOI":"10.1007\/s00220-006-1526-7","volume":"264","author":"S Friedlander","year":"2006","unstructured":"Friedlander, S., Pavlovi\u0107, N., Shvydkoy, R.: Nonlinear instability for the Navier\u2013Stokes equations. Commun. Math. Phys. 264(2), 335\u2013347 (2006)","journal-title":"Commun. Math. Phys."},{"issue":"2","key":"255_CR17","doi-asserted-by":"publisher","first-page":"198","DOI":"10.1063\/1.165829","volume":"1","author":"S Friedlander","year":"1991","unstructured":"Friedlander, S., Vishik, M.M.: Dynamo theory, vorticity generation, and exponential stretching. Chaos 1(2), 198\u2013205 (1991)","journal-title":"Chaos"},{"issue":"19","key":"255_CR18","doi-asserted-by":"publisher","first-page":"4365","DOI":"10.1103\/PhysRevLett.84.4365","volume":"84","author":"A Gailitis","year":"2000","unstructured":"Gailitis, A., Lielausis, O., Dement\u2019ev, S., Placatis, E., Cifersons, A., Gerbeth, G., Gundrum, T., Stefani, F., Christen, M., Hanel, H., Will, G.: Detection of a flow induced magnetic field eigenmode in the Riga dynamo facility. Phys. Rev. Lett. 84(19), 4365\u20134368 (2000)","journal-title":"Phys. Rev. Lett."},{"key":"255_CR19","unstructured":"Gardner, V., Liss, K.L., Mattingly, J.C.: A pathwise approach to the enhanced dissipation of passive scalars advected by shear flows (2024)"},{"issue":"3","key":"255_CR20","doi-asserted-by":"publisher","first-page":"815","DOI":"10.1137\/S0036141004444603","volume":"37","author":"D G\u00e9rard-Varet","year":"2005","unstructured":"G\u00e9rard-Varet, D.: Oscillating solutions of incompressible magnetohydrodynamics and dynamo effect. SIAM J. Math. Anal. 37(3), 815\u2013840 (2005)","journal-title":"SIAM J. Math. Anal."},{"issue":"5","key":"255_CR21","doi-asserted-by":"publisher","first-page":"677","DOI":"10.1016\/j.anihpc.2006.04.005","volume":"24","author":"D G\u00e9rard-Varet","year":"2007","unstructured":"G\u00e9rard-Varet, D., Rousset, F.: Shear layer solutions of incompressible MHD and dynamo effect. Ann. Inst. H. Poincar\u00e9 C Anal. Non Lin\u00e9aire 24(5), 677\u2013710 (2007)","journal-title":"Ann. Inst. H. Poincar\u00e9 C Anal. Non Lin\u00e9aire"},{"key":"255_CR22","volume-title":"Mathematical Methods for the Magnetohydrodynamics of Liquid Metals","author":"J-F Gerbeau","year":"2007","unstructured":"Gerbeau, J.-F., Le Bris, C., Leli\u00e9vre, T.: Mathematical Methods for the Magnetohydrodynamics of Liquid Metals. Oxford University Press (2007)"},{"key":"255_CR23","series-title":"Applied Mathematical Sciences","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-1052-8","volume-title":"Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory, and Climate Dynamics","author":"M Ghil","year":"1987","unstructured":"Ghil, M., Childress, S.: Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory, and Climate Dynamics. Applied Mathematical Sciences, vol. 60, 1st edn. Springer, New York (1987)","edition":"1"},{"issue":"1\u20134","key":"255_CR24","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1080\/03091928808208888","volume":"44","author":"AD Gilbert","year":"1988","unstructured":"Gilbert, A.D.: Fast dynamo action in the Ponomarenko dynamo. Geophys. Astrophys. Fluid Dyn. 44(1\u20134), 241\u2013258 (1988)","journal-title":"Geophys. Astrophys. Fluid Dyn."},{"issue":"9","key":"255_CR25","doi-asserted-by":"publisher","first-page":"1067","DOI":"10.1002\/1097-0312(200009)53:9<1067::AID-CPA1>3.0.CO;2-Q","volume":"53","author":"E Grenier","year":"2000","unstructured":"Grenier, E.: On the nonlinear instability of Euler and Prandtl equations. Commun. Pure Appl. Math. 53(9), 1067\u20131091 (2000)","journal-title":"Commun. Pure Appl. Math."},{"key":"255_CR26","doi-asserted-by":"crossref","unstructured":"Grenier, E., Guo, Y., Nguyen, T.T.: On the spectral instability of parallel shear flows, S\u00e9minaire Laurent Schwartz-\u00e9quations aux d\u00e9riv\u00e9es partielles et applications. Ann\u00e9e 2014\u20132015, 2016, pp. Exp. No. XXII, 14","DOI":"10.5802\/slsedp.82"},{"issue":"16","key":"255_CR27","doi-asserted-by":"publisher","first-page":"3085","DOI":"10.1215\/00127094-3645437","volume":"165","author":"E Grenier","year":"2016","unstructured":"Grenier, E., Guo, Y., Nguyen, T.T.: Spectral instability of characteristic boundary layer flows. Duke Math. J. 165(16), 3085\u20133146 (2016)","journal-title":"Duke Math. J."},{"key":"255_CR28","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-53393-8","volume-title":"Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften","author":"T Kato","year":"1966","unstructured":"Kato, T.: Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, vol. Band 132. Springer, New York (1966)"},{"issue":"3","key":"255_CR29","doi-asserted-by":"publisher","first-page":"623","DOI":"10.1007\/BF02101659","volume":"173","author":"I Klapper","year":"1995","unstructured":"Klapper, I., Young, L.-S.: Rigorous bounds on the fast dynamo growth rate involving topological entropy. Commun. Math. Phys. 173(3), 623\u2013646 (1995)","journal-title":"Commun. Math. Phys."},{"key":"255_CR30","first-page":"412","volume":"85","author":"J Larmor","year":"1919","unstructured":"Larmor, J.: Possible rotational origin of magnetic fields of Sun and Earth. Electr. Rev. 85, 412 (1919)","journal-title":"Electr. Rev."},{"key":"255_CR31","unstructured":"Lebedev, N.N.: Special Functions and Their Applications, Revised (R. A. Silverman, ed.), Dover Publications, Inc., New York (1972)"},{"key":"255_CR32","volume-title":"Magnetic Field Generation in Electrically Conducting Fluids","author":"HK Moffatt","year":"1978","unstructured":"Moffatt, H.K.: Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press (1978)"},{"key":"255_CR33","doi-asserted-by":"crossref","unstructured":"Navarro-Fern\u00e1ndez, V., Villringer, D.: Nonlinear instability for the 3D MHD equations around the Taylor\u2013 Couette flow (2025)","DOI":"10.1088\/1361-6544\/ae65fc"},{"issue":"6","key":"255_CR34","doi-asserted-by":"publisher","first-page":"1606","DOI":"10.1063\/1.1571547","volume":"15","author":"C Normand","year":"2003","unstructured":"Normand, C.: Ponomarenko dynamo with time-periodic flow. Phys. Fluids 15(6), 1606\u20131611 (2003)","journal-title":"Phys. Fluids"},{"key":"255_CR35","unstructured":"Olver, F.W.J., Daalhuis, A.B.O., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W., Mille, B.R., Saunders, B.V., Cohl, H.S., McClain, M.A.: NIST Digital Library of Mathematical Functions. http:\/\/dlmf.nist.gov\/ (2020)"},{"key":"255_CR36","volume-title":"Asymptotics and Special Functions","author":"FWJ Olver","year":"1974","unstructured":"Olver, F.W.J.: Asymptotics and Special Functions. Elsevier Academic Press (1974)"},{"issue":"12","key":"255_CR37","doi-asserted-by":"publisher","DOI":"10.1063\/1.3035911","volume":"15","author":"M Peyrot","year":"2008","unstructured":"Peyrot, M., Gilbert, A., Plunian, F.: Oscillating Ponomarenko dynamo in the highly conducting limit. Phys. Plasmas 15(12), 122104 (2008)","journal-title":"Phys. Plasmas"},{"key":"255_CR38","doi-asserted-by":"publisher","first-page":"775","DOI":"10.1007\/BF00853190","volume":"14","author":"YB Ponomarenko","year":"1973","unstructured":"Ponomarenko, Y.B.: Theory of the hydromagnetic generator. J. Appl. Mech. Tech. Phys. 14, 775\u2013778 (1973)","journal-title":"J. Appl. Mech. Tech. Phys."},{"key":"255_CR39","doi-asserted-by":"publisher","first-page":"535","DOI":"10.1098\/rsta.1970.0011","volume":"266","author":"GO Roberts","year":"1970","unstructured":"Roberts, G.O.: Spatially periodic dynamos. Philos. Trans. R. Soc. Lond. A 266, 535\u2013558 (1970)","journal-title":"Philos. Trans. R. Soc. Lond. A"},{"key":"255_CR40","volume-title":"An Introduction to Magnetohydrodynamics","author":"PH Roberts","year":"1967","unstructured":"Roberts, P.H.: An Introduction to Magnetohydrodynamics. Longmans, London (1967)"},{"key":"255_CR41","doi-asserted-by":"crossref","unstructured":"Rowan, K.: A subsequentially fast dynamo on $$\\mathbb{t}^3$$. arXiv:2505.23936 (2025)","DOI":"10.1007\/s00205-026-02255-8"},{"key":"255_CR42","volume-title":"The Magnetic Universe: Geophysical and Astrophysical Dynamo Theory","author":"G R\u00fcdiger","year":"2006","unstructured":"R\u00fcdiger, G., Hollerbach, R.: The Magnetic Universe: Geophysical and Astrophysical Dynamo Theory. Wiley (2006)"},{"key":"255_CR43","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1017\/S0022112088003167","volume":"197","author":"A Ruzmaikin","year":"1988","unstructured":"Ruzmaikin, A., Sokoloff, D., Shukurov, A.: Hydromagnetic screw dynamo. J. Fluid Mech. 197, 39\u201356 (1988)","journal-title":"J. Fluid Mech."},{"issue":"5","key":"255_CR44","doi-asserted-by":"publisher","first-page":"635","DOI":"10.1002\/cpa.3160360506","volume":"36","author":"M Sermange","year":"1983","unstructured":"Sermange, M., Temam, R.: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36(5), 635\u2013664 (1983)","journal-title":"Commun. Pure Appl. Math."},{"key":"255_CR45","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1017\/S0022112087001800","volume":"180","author":"AM Soward","year":"1987","unstructured":"Soward, A.M.: Fast dynamo action in a steady flow. J. Fluid Mech. 180, 267\u2013295 (1987)","journal-title":"J. Fluid Mech."},{"issue":"2","key":"255_CR46","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1070\/PU1972v015n02ABEH004960","volume":"15","author":"SI Vainshtein","year":"1972","unstructured":"Vainshtein, S.I., Zeldovich, Y.B.: Origin of magnetic fields in astrophysics (turbulent \u201cdynamo\u2019\u2019 mechanisms). Soviet Phys. Uspekhi 15(2), 159 (1972)","journal-title":"Soviet Phys. Uspekhi"},{"key":"255_CR47","doi-asserted-by":"publisher","first-page":"543","DOI":"10.1007\/s11214-009-9546-1","volume":"152","author":"G Verhille","year":"2010","unstructured":"Verhille, G., Plihon, N., Bourgoin, M., Odier, P., Pinton, J.-F.: Laboratory dynamo experiments. Space Sci. Rev. 152, 543\u2013564 (2010)","journal-title":"Space Sci. Rev."},{"key":"255_CR48","doi-asserted-by":"publisher","first-page":"26","DOI":"10.1214\/25-ECP676","volume":"30","author":"D Villringer","year":"2025","unstructured":"Villringer, D.: Enhanced dissipation via the Malliavin calculus. Electron. Commun. Probab. 30, 26 (2025)","journal-title":"Electron. Commun. Probab."},{"key":"255_CR49","unstructured":"Vishik, M.M.: The periodic dynamo, Mathematical methods in seismology and geodynamics, No. 19 , (1986), pp. 186\u2013215, 221, 224 (Russian)"},{"issue":"1\u20133","key":"255_CR50","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1080\/03091928908219531","volume":"48","author":"MM Vishik","year":"1989","unstructured":"Vishik, M.M.: Magnetic field generation by the motion of a highly conducting fluid. Geophys. Astrophys. Fluid Dyn. 48(1\u20133), 151\u2013167 (1989)","journal-title":"Geophys. Astrophys. Fluid Dyn."},{"issue":"3","key":"255_CR51","doi-asserted-by":"publisher","first-page":"507","DOI":"10.1007\/s11425-018-9461-8","volume":"64","author":"D Wei","year":"2021","unstructured":"Wei, D.: Diffusion and mixing in fluid flow via the resolvent estimate. Sci China Math 64(3), 507\u2013518 (2021)","journal-title":"Sci China Math"},{"key":"255_CR52","unstructured":"Wynne, J.: The smooth Ponomarenko dynamo. PhD thesis (2017)"},{"key":"255_CR53","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/S0022112084001488","volume":"144","author":"YB Zeldovich","year":"1984","unstructured":"Zeldovich, Y.B., Ruzmaikin, A.A., Molchanov, S.A., Sokoloff, D.D.: Kinematic dynamo problem in a linear velocity field. J. Fluid Mech. 144, 1\u201311 (1984)","journal-title":"J. Fluid Mech."},{"key":"255_CR54","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1515\/9781400862979.93","volume":"I","author":"YB Zeldovich","year":"1992","unstructured":"Zeldovich, Y.B.: A magnetic field in the two-dimensional motion of a conducting turbulent fluid. Selected Works of Yakov Borisovich Zeldovich I, 93\u201396 (1992)","journal-title":"Selected Works of Yakov Borisovich Zeldovich"},{"key":"255_CR55","first-page":"980","volume":"78","author":"YB Zeldovich","year":"1980","unstructured":"Zeldovich, Y.B., Ruzmaikin, A.A.: Magnetic field of a conducting fluid in two-dimensional motion. Zh. Eksp. Teor. Fiz. 78, 980\u2013986 (1980)","journal-title":"Zh. Eksp. Teor. Fiz."}],"container-title":["Annals of PDE"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40818-026-00255-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40818-026-00255-1","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40818-026-00255-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T11:02:56Z","timestamp":1791543776000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40818-026-00255-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,10,9]]},"references-count":55,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,12]]}},"alternative-id":["255"],"URL":"https:\/\/doi.org\/10.1007\/s40818-026-00255-1","relation":{},"ISSN":["2524-5317","2199-2576"],"issn-type":[{"value":"2524-5317","type":"print"},{"value":"2199-2576","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,10,9]]},"assertion":[{"value":"8 October 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 September 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 October 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"32"}}