{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,16]],"date-time":"2025-10-16T00:41:14Z","timestamp":1760575274699,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:00:00Z","timestamp":1760140800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Nanyang Normal University Doctoral Research Startup Fund Project in Natural Sciences","award":["2020ZX001"],"award-info":[{"award-number":["2020ZX001"]}]},{"name":"Program for Science and Technology Innovation Talents in Universities of Henan Province","award":["25HASTIT024"],"award-info":[{"award-number":["25HASTIT024"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this article, we establish the existence of global mild solutions to the Vlasov\u2013Fokker\u2013Planck equation with local alignment forces under specular reflection boundary conditions in the low-regularity function space Lk1LT\u221eLv2. A key difficulty is that the macroscopic averaged velocity u does not directly possess a dissipative structure in the equation. To overcome this, we rely on the dissipation u\u2212b from the linear part, combined with the dissipation of the macroscopic component b derived from the associated macroscopic equation. Moreover, since no direct energy functional is available for u, we fully exploit the dissipative mechanisms of both u\u2212b and b when handling the estimates for the nonlinear terms.<\/jats:p>","DOI":"10.3390\/axioms14100760","type":"journal-article","created":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T11:57:42Z","timestamp":1760529462000},"page":"760","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Global Solutions to the Vlasov\u2013Fokker\u2013Planck Equation with Local Alignment Forces Under Specular Reflection Boundary Condition"],"prefix":"10.3390","volume":"14","author":[{"given":"Yanming","family":"Chang","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Nanyang Normal University, Nanyang 473061, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yingzhe","family":"Fan","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Nanyang Normal University, Nanyang 473061, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,10,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1081","DOI":"10.1512\/iumj.2004.53.2574","article-title":"The Boltzmann equation in the whole space","volume":"53","author":"Guo","year":"2004","journal-title":"Indiana Univ. Math. J."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"932","DOI":"10.1002\/cpa.21920","article-title":"Global Mild Solutions of the Landau and Non-Cutoff Boltzmann Equations","volume":"74","author":"Duan","year":"2021","journal-title":"Comm. Pure Appl. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1887","DOI":"10.1088\/0951-7715\/29\/7\/1887","article-title":"Global classical solutions of the Vlasov-Fokker-Planck equation with local alignment forces","volume":"29","author":"Choi","year":"2016","journal-title":"Nonlinearity"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1137\/120866828","article-title":"Existence of weak solutions to kinetic flocking models","volume":"45","author":"Karper","year":"2013","journal-title":"SIAM J. Math. Anal."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1142\/S0218202515500050","article-title":"Hydrodynamic limit of the kinetic Cucker-Smale flocking model","volume":"25","author":"Karper","year":"2015","journal-title":"Math. Models Methods Appl. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"335","DOI":"10.3934\/krm.2015.8.335","article-title":"Compressible Euler equations interacting with incompressible flow","volume":"8","author":"Choi","year":"2015","journal-title":"Kinet. Relat. Models"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1104","DOI":"10.1002\/cpa.10040","article-title":"The Vlasov-Poisson-Boltzmann system near Maxwellians","volume":"55","author":"Guo","year":"2002","journal-title":"Comm. Pure Appl. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1090\/S0894-0347-2011-00722-4","article-title":"The Vlasov-Poisson-Laudau system in a periodic box","volume":"25","author":"Guo","year":"2012","journal-title":"J. Am. Math. Soc."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"687","DOI":"10.3934\/krm.2013.6.687","article-title":"Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force","volume":"6","author":"Duan","year":"2013","journal-title":"Kinet. Relat. Models"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"876","DOI":"10.1016\/S0252-9602(15)30026-6","article-title":"Time-periodic solutions of the Vlasov-Poisson-Fokker-Planck system","volume":"35","author":"Duan","year":"2015","journal-title":"Acta Math. Sci."},{"key":"ref_11","first-page":"681","article-title":"On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian","volume":"18","author":"Hwang","year":"2013","journal-title":"Discrete Contin. Dyn. Syst. Ser. B"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1006\/jdeq.1995.1146","article-title":"Smoothing effect for the non-linear Vlasov-Poisson-Fokker-Planck system","volume":"122","author":"Bouchut","year":"1995","journal-title":"J. Differ. Equ."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1016\/j.anihpc.2014.10.002","article-title":"On the analysis of a coupled kinetic-fluid model with local alignment forces","volume":"33","author":"Carrillo","year":"2016","journal-title":"Ann. Inst. Henri Poincare"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1142\/S0218202516500068","article-title":"Critical thresholds in 1D Euler equations with non-local forces","volume":"26","author":"Carrillo","year":"2016","journal-title":"Math. Model. Methods Appl. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"227","DOI":"10.3934\/krm.2011.4.227","article-title":"Global classical solutions close to equilibrium to the Vlasov-Fokker-Planck-Euler system","volume":"4","author":"Carrillo","year":"2011","journal-title":"Kinet. Relat. Models"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1137\/090757290","article-title":"Asymptotic flocking dynamics for the kinetic Cucker-Smale model","volume":"42","author":"Carrillo","year":"2010","journal-title":"SIAM J. Math. Anal."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1007\/s002050050038","article-title":"Dispersion relations for the linearized Fokker-Planck equation","volume":"138","author":"Degond","year":"1997","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/s00205-008-0184-7","article-title":"Stability of the front under a Vlasov-Fokker-Planck dynamics","volume":"195","author":"Esposito","year":"2010","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Villani, C. (2002). A review of mathematical topics in collisional kinetic theory. Handbook of Mathematical Fluid Dynamics, Elsevier.","DOI":"10.1016\/S1874-5792(02)80004-0"},{"key":"ref_20","unstructured":"Frank, T.-D. (2005). Nonlinear Fokker-Planck Equations: Fundamentals and Applications, Springer."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1007\/s00220-004-1278-1","article-title":"A Half-space Problem for the Boltzmann Equation with Specular Reflection Boundary Condition","volume":"255","author":"Yang","year":"2005","journal-title":"Commun. Math. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/s10955-023-03203-6","article-title":"The Boltzmann equation with a class of large-amplitude initial data and specular reflection boundary condition","volume":"190","author":"Duan","year":"2023","journal-title":"J. Stat. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1389","DOI":"10.1007\/s00205-020-01496-5","article-title":"The Landau Equation with the Specular Reflection Boundary Condition","volume":"236","author":"Guo","year":"2020","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1007\/s00205-022-01818-9","article-title":"The Vlasov-Poisson-Landau System with the Specular-Reflection Boundary Condition","volume":"246","author":"Dong","year":"2022","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"031504","DOI":"10.1063\/5.0132586","article-title":"Global solutions and exponential time decay rates to the Navier-Stokes-Vlasov-Fokker-Planck system in low regularity space","volume":"64","author":"Tan","year":"2023","journal-title":"J. Math. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"021503","DOI":"10.1063\/5.0185948","article-title":"The Fokker-Planck-Boltzmann equation in the finite channel","volume":"65","author":"Lei","year":"2024","journal-title":"J. Math. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2193","DOI":"10.1088\/1361-6544\/acc3f0","article-title":"Low Regularity Solutions for the Vlasov-Poisson-Landau\/Boltzmann System","volume":"36","author":"Deng","year":"2023","journal-title":"Nonlinearity"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/10\/760\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T12:09:43Z","timestamp":1760530183000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/10\/760"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,11]]},"references-count":27,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2025,10]]}},"alternative-id":["axioms14100760"],"URL":"https:\/\/doi.org\/10.3390\/axioms14100760","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,11]]}}}