[{"acknowledgement":"The first author has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 948819) Image 1. The second author is supported by the VICI subsidy VI.C.212.027 of the Netherlands Organisation for Scientific Research (NWO).","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","quality_controlled":"1","oa_version":"Published Version","project":[{"grant_number":"948819","_id":"0aa76401-070f-11eb-9043-b5bb049fa26d","name":"Bridging Scales in Random Materials","call_identifier":"H2020"}],"_id":"13135","publication_identifier":{"issn":["0022-0396"],"eissn":["1090-2732"]},"date_updated":"2024-01-29T11:04:41Z","oa":1,"volume":368,"article_processing_charge":"Yes (in subscription journal)","author":[{"id":"673cd0cc-9b9a-11eb-b144-88f30e1fbb72","first_name":"Antonio","orcid":"0000-0002-9573-2962","full_name":"Agresti, Antonio","last_name":"Agresti"},{"first_name":"Mark","full_name":"Veraar, Mark","last_name":"Veraar"}],"abstract":[{"lang":"eng","text":"In this paper we consider a class of stochastic reaction-diffusion equations. We provide local well-posedness, regularity, blow-up criteria and positivity of solutions. The key novelties of this work are related to the use transport noise, critical spaces and the proof of higher order regularity of solutions – even in case of non-smooth initial data. Crucial tools are Lp(Lp)-theory, maximal regularity estimates and sharp blow-up criteria. We view the results of this paper as a general toolbox for establishing global well-posedness for a large class of reaction-diffusion systems of practical interest, of which many are completely open. In our follow-up work [8], the results of this paper are applied in the specific cases of the Lotka-Volterra equations and the Brusselator model."}],"publication_status":"published","citation":{"ieee":"A. Agresti and M. Veraar, “Reaction-diffusion equations with transport noise and critical superlinear diffusion: Local well-posedness and positivity,” <i>Journal of Differential Equations</i>, vol. 368, no. 9. Elsevier, pp. 247–300, 2023.","apa":"Agresti, A., &#38; Veraar, M. (2023). Reaction-diffusion equations with transport noise and critical superlinear diffusion: Local well-posedness and positivity. <i>Journal of Differential Equations</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.jde.2023.05.038\">https://doi.org/10.1016/j.jde.2023.05.038</a>","chicago":"Agresti, Antonio, and Mark Veraar. “Reaction-Diffusion Equations with Transport Noise and Critical Superlinear Diffusion: Local Well-Posedness and Positivity.” <i>Journal of Differential Equations</i>. Elsevier, 2023. <a href=\"https://doi.org/10.1016/j.jde.2023.05.038\">https://doi.org/10.1016/j.jde.2023.05.038</a>.","mla":"Agresti, Antonio, and Mark Veraar. “Reaction-Diffusion Equations with Transport Noise and Critical Superlinear Diffusion: Local Well-Posedness and Positivity.” <i>Journal of Differential Equations</i>, vol. 368, no. 9, Elsevier, 2023, pp. 247–300, doi:<a href=\"https://doi.org/10.1016/j.jde.2023.05.038\">10.1016/j.jde.2023.05.038</a>.","ama":"Agresti A, Veraar M. Reaction-diffusion equations with transport noise and critical superlinear diffusion: Local well-posedness and positivity. <i>Journal of Differential Equations</i>. 2023;368(9):247-300. doi:<a href=\"https://doi.org/10.1016/j.jde.2023.05.038\">10.1016/j.jde.2023.05.038</a>","ista":"Agresti A, Veraar M. 2023. Reaction-diffusion equations with transport noise and critical superlinear diffusion: Local well-posedness and positivity. Journal of Differential Equations. 368(9), 247–300.","short":"A. Agresti, M. Veraar, Journal of Differential Equations 368 (2023) 247–300."},"ddc":["510"],"isi":1,"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"external_id":{"isi":["001019018700001"]},"title":"Reaction-diffusion equations with transport noise and critical superlinear diffusion: Local well-posedness and positivity","ec_funded":1,"year":"2023","doi":"10.1016/j.jde.2023.05.038","page":"247-300","file_date_updated":"2024-01-29T11:03:09Z","issue":"9","publication":"Journal of Differential Equations","status":"public","intvolume":"       368","type":"journal_article","day":"25","file":[{"date_updated":"2024-01-29T11:03:09Z","access_level":"open_access","file_name":"2023_JourDifferentialEquations_Agresti.pdf","file_size":834638,"date_created":"2024-01-29T11:03:09Z","checksum":"246b703b091dfe047bfc79abf0891a63","content_type":"application/pdf","relation":"main_file","creator":"dernst","file_id":"14895","success":1}],"date_created":"2023-06-18T22:00:45Z","department":[{"_id":"JuFi"}],"has_accepted_license":"1","language":[{"iso":"eng"}],"publisher":"Elsevier","scopus_import":"1","date_published":"2023-09-25T00:00:00Z","article_type":"original","month":"09"},{"language":[{"iso":"eng"}],"scopus_import":"1","publisher":"Elsevier","date_published":"2021-05-25T00:00:00Z","article_type":"original","month":"05","date_created":"2021-03-14T23:01:32Z","file":[{"relation":"main_file","content_type":"application/pdf","creator":"dernst","file_id":"9267","success":1,"access_level":"open_access","date_updated":"2021-03-22T07:18:01Z","file_size":473310,"file_name":"2021_JourDiffEquations_Cornalba.pdf","checksum":"c630b691fb9e716b02aa6103a9794ec8","date_created":"2021-03-22T07:18:01Z"}],"department":[{"_id":"JuFi"}],"has_accepted_license":"1","status":"public","intvolume":"       284","type":"journal_article","day":"25","page":"253-283","file_date_updated":"2021-03-22T07:18:01Z","publication":"Journal of Differential Equations","issue":"5","external_id":{"isi":["000634823300010"]},"title":"Well-posedness for a regularised inertial Dean–Kawasaki model for slender particles in several space dimensions","ec_funded":1,"year":"2021","doi":"10.1016/j.jde.2021.02.048","ddc":["510"],"isi":1,"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"author":[{"id":"2CEB641C-A400-11E9-A717-D712E6697425","full_name":"Cornalba, Federico","last_name":"Cornalba","first_name":"Federico"},{"last_name":"Shardlow","full_name":"Shardlow, Tony","first_name":"Tony"},{"first_name":"Johannes","last_name":"Zimmer","full_name":"Zimmer, Johannes"}],"abstract":[{"lang":"eng","text":"A stochastic PDE, describing mesoscopic fluctuations in systems of weakly interacting inertial particles of finite volume, is proposed and analysed in any finite dimension . It is a regularised and inertial version of the Dean–Kawasaki model. A high-probability well-posedness theory for this model is developed. This theory improves significantly on the spatial scaling restrictions imposed in an earlier work of the same authors, which applied only to significantly larger particles in one dimension. The well-posedness theory now applies in d-dimensions when the particle-width ϵ is proportional to  for  and N is the number of particles. This scaling is optimal in a certain Sobolev norm. Key tools of the analysis are fractional Sobolev spaces, sharp bounds on Bessel functions, separability of the regularisation in the d-spatial dimensions, and use of the Faà di Bruno's formula."}],"citation":{"apa":"Cornalba, F., Shardlow, T., &#38; Zimmer, J. (2021). Well-posedness for a regularised inertial Dean–Kawasaki model for slender particles in several space dimensions. <i>Journal of Differential Equations</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.jde.2021.02.048\">https://doi.org/10.1016/j.jde.2021.02.048</a>","ieee":"F. Cornalba, T. Shardlow, and J. Zimmer, “Well-posedness for a regularised inertial Dean–Kawasaki model for slender particles in several space dimensions,” <i>Journal of Differential Equations</i>, vol. 284, no. 5. Elsevier, pp. 253–283, 2021.","chicago":"Cornalba, Federico, Tony Shardlow, and Johannes Zimmer. “Well-Posedness for a Regularised Inertial Dean–Kawasaki Model for Slender Particles in Several Space Dimensions.” <i>Journal of Differential Equations</i>. Elsevier, 2021. <a href=\"https://doi.org/10.1016/j.jde.2021.02.048\">https://doi.org/10.1016/j.jde.2021.02.048</a>.","ama":"Cornalba F, Shardlow T, Zimmer J. Well-posedness for a regularised inertial Dean–Kawasaki model for slender particles in several space dimensions. <i>Journal of Differential Equations</i>. 2021;284(5):253-283. doi:<a href=\"https://doi.org/10.1016/j.jde.2021.02.048\">10.1016/j.jde.2021.02.048</a>","mla":"Cornalba, Federico, et al. “Well-Posedness for a Regularised Inertial Dean–Kawasaki Model for Slender Particles in Several Space Dimensions.” <i>Journal of Differential Equations</i>, vol. 284, no. 5, Elsevier, 2021, pp. 253–83, doi:<a href=\"https://doi.org/10.1016/j.jde.2021.02.048\">10.1016/j.jde.2021.02.048</a>.","short":"F. Cornalba, T. Shardlow, J. Zimmer, Journal of Differential Equations 284 (2021) 253–283.","ista":"Cornalba F, Shardlow T, Zimmer J. 2021. Well-posedness for a regularised inertial Dean–Kawasaki model for slender particles in several space dimensions. Journal of Differential Equations. 284(5), 253–283."},"publication_status":"published","project":[{"grant_number":"754411","call_identifier":"H2020","name":"ISTplus - Postdoctoral Fellowships","_id":"260C2330-B435-11E9-9278-68D0E5697425"}],"oa_version":"Published Version","quality_controlled":"1","acknowledgement":"All authors thank the anonymous referee for his/her careful reading of the manuscript and valuable suggestions. This paper was motivated by stimulating discussions at the First Berlin–Leipzig Workshop on Fluctuating Hydrodynamics in August 2019 with Ana Djurdjevac, Rupert Klein and Ralf Kornhuber. JZ gratefully acknowledges funding by a Royal Society Wolfson Research Merit Award. FC gratefully acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411.","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","publication_identifier":{"eissn":["1090-2732"],"issn":["0022-0396"]},"_id":"9240","article_processing_charge":"Yes (via OA deal)","volume":284,"oa":1,"date_updated":"2023-08-07T14:08:05Z"},{"_id":"8691","publication_identifier":{"issn":["0022-0396"]},"extern":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","quality_controlled":"1","oa_version":"Preprint","arxiv":1,"date_updated":"2021-01-12T08:20:33Z","volume":269,"oa":1,"article_processing_charge":"No","abstract":[{"text":"Given l>2ν>2d≥4, we prove the persistence of a Cantor--family of KAM tori of measure O(ε1/2−ν/l) for any non--degenerate nearly integrable Hamiltonian system of class Cl(D×Td), where D⊂Rd is a bounded domain, provided that the size ε of the perturbation is sufficiently small. This extends a result by D. Salamon in \\cite{salamon2004kolmogorov} according to which we do have the persistence of a single KAM torus in the same framework. Moreover, it is well--known that, for the persistence of a single torus, the regularity assumption can not be improved.","lang":"eng"}],"author":[{"id":"52DF3E68-AEFA-11EA-95A4-124A3DDC885E","first_name":"Edmond","full_name":"Koudjinan, Edmond","last_name":"Koudjinan","orcid":"0000-0003-2640-4049"}],"keyword":["Analysis"],"publication_status":"published","citation":{"ista":"Koudjinan E. 2020. A KAM theorem for finitely differentiable Hamiltonian systems. Journal of Differential Equations. 269(6), 4720–4750.","short":"E. Koudjinan, Journal of Differential Equations 269 (2020) 4720–4750.","mla":"Koudjinan, Edmond. “A KAM Theorem for Finitely Differentiable Hamiltonian Systems.” <i>Journal of Differential Equations</i>, vol. 269, no. 6, Elsevier, 2020, pp. 4720–50, doi:<a href=\"https://doi.org/10.1016/j.jde.2020.03.044\">10.1016/j.jde.2020.03.044</a>.","ama":"Koudjinan E. A KAM theorem for finitely differentiable Hamiltonian systems. <i>Journal of Differential Equations</i>. 2020;269(6):4720-4750. doi:<a href=\"https://doi.org/10.1016/j.jde.2020.03.044\">10.1016/j.jde.2020.03.044</a>","chicago":"Koudjinan, Edmond. “A KAM Theorem for Finitely Differentiable Hamiltonian Systems.” <i>Journal of Differential Equations</i>. Elsevier, 2020. <a href=\"https://doi.org/10.1016/j.jde.2020.03.044\">https://doi.org/10.1016/j.jde.2020.03.044</a>.","ieee":"E. Koudjinan, “A KAM theorem for finitely differentiable Hamiltonian systems,” <i>Journal of Differential Equations</i>, vol. 269, no. 6. Elsevier, pp. 4720–4750, 2020.","apa":"Koudjinan, E. (2020). A KAM theorem for finitely differentiable Hamiltonian systems. <i>Journal of Differential Equations</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.jde.2020.03.044\">https://doi.org/10.1016/j.jde.2020.03.044</a>"},"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1909.04099"}],"title":"A KAM theorem for finitely differentiable Hamiltonian systems","external_id":{"arxiv":["1909.04099"]},"doi":"10.1016/j.jde.2020.03.044","year":"2020","issue":"6","publication":"Journal of Differential Equations","page":"4720-4750","intvolume":"       269","status":"public","day":"05","type":"journal_article","date_created":"2020-10-21T15:03:05Z","publisher":"Elsevier","language":[{"iso":"eng"}],"month":"09","article_type":"original","date_published":"2020-09-05T00:00:00Z"}]