{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,18]],"date-time":"2025-10-18T10:53:44Z","timestamp":1760784824532,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,4,2]],"date-time":"2020-04-02T00:00:00Z","timestamp":1585785600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper is devoted to the derivation and mathematical analysis of new thermostatted kinetic theory frameworks for the modeling of nonequilibrium complex systems composed by particles whose microscopic state includes a vectorial state variable. The mathematical analysis refers to the global existence and uniqueness of the solution of the related Cauchy problem. Specifically, the paper is divided in two parts. In the first part the thermostatted framework with a continuous vectorial variable is proposed and analyzed. The framework consists of a system of partial integro-differential equations with quadratic type nonlinearities. In the second part the thermostatted framework with a discrete vectorial variable is investigated. Real world applications, such as social systems and crowd dynamics, and future research directions are outlined in the paper.<\/jats:p>","DOI":"10.3390\/sym12040517","type":"journal-article","created":{"date-parts":[[2020,4,2]],"date-time":"2020-04-02T13:39:34Z","timestamp":1585834774000},"page":"517","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["On the Cauchy Problem of Vectorial Thermostatted Kinetic Frameworks"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1122-7510","authenticated-orcid":false,"given":"Carlo","family":"Bianca","sequence":"first","affiliation":[{"name":"Laboratoire Quartz EA 7393, \u00c9cole Sup\u00e9rieure d\u2019Ing\u00e9nieurs en G\u00e9nie \u00c9lectrique, Productique et Management Industriel, 95092 Cergy Pontoise CEDEX, France"},{"name":"Laboratoire de Recherche en Eco-Innovation Industrielle et Energ\u00e9tique, \u00c9cole Sup\u00e9rieure d\u2019Ing\u00e9nieurs en G\u00e9nie \u00c9lectrique, Productique et Management Industriel, 95092 Cergy Pontoise CEDEX, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bruno","family":"Carbonaro","sequence":"additional","affiliation":[{"name":"Dipartimento di Matematica e Fisica, Universit\u00e0 degli Studi della Campania \u201cL. 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