{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:20:04Z","timestamp":1760059204589,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,5,27]],"date-time":"2025-05-27T00:00:00Z","timestamp":1748304000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Shanghai","award":["20ZR1419400","MIT\/DRG10\/24-25"],"award-info":[{"award-number":["20ZR1419400","MIT\/DRG10\/24-25"]}]},{"name":"Education University of Hong Kong","award":["20ZR1419400","MIT\/DRG10\/24-25"],"award-info":[{"award-number":["20ZR1419400","MIT\/DRG10\/24-25"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The Euler\u2013Boltzmann equations are an important class of mathematical models that describe the coupling between particle transport and macroscopic fluid dynamics. They find broad applications in plasma physics, rarefied gas dynamics, and astrophysics. In these fields, incorporating a time-dependent damping term is crucial for modeling real-world scenarios, as opposed to idealized inviscid conditions. In recent years, there has been growing interest in the long-time behavior of their solutions. This paper focuses on the initial value problem for the three-dimensional Euler\u2013Boltzmann equations with time-dependent damping, aiming to investigate the finite-time blowup behavior of classical solutions. We use an integration method with general test function f and show that if the initial data are sufficiently large, classical solutions of the Euler\u2013Boltzmann equations with time-dependent damping in R3 will blowup on or before the finite time T*&gt;0.<\/jats:p>","DOI":"10.3390\/sym17060835","type":"journal-article","created":{"date-parts":[[2025,5,29]],"date-time":"2025-05-29T04:46:38Z","timestamp":1748493998000},"page":"835","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Singularity Formation of Classical Solutions to Euler\u2013Boltzmann Equations with Damping in R3"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6195-8453","authenticated-orcid":false,"given":"Jianli","family":"Liu","sequence":"first","affiliation":[{"name":"Department of Mathematics, Shanghai University, Shanghai 200444, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mengyan","family":"Liu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Shanghai University, Shanghai 200444, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5035-3555","authenticated-orcid":false,"given":"Manwai","family":"Yuen","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Information Technology, The Education University of Hong Kong, Hong Kong, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,27]]},"reference":[{"key":"ref_1","unstructured":"Mihalas, O., and Mihalas, B. (1984). Foundations of Radiation Hydrodynamics, Oxford University Press."},{"key":"ref_2","unstructured":"Pomraning, G. (1973). The Equations of Radiation Hydrodynamics, Pergamon."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1055","DOI":"10.1115\/1.3607836","article-title":"Physics of Shock Waves and High-Temperture Hydrodynamic Phenomenon","volume":"34","author":"Zeldovich","year":"1967","journal-title":"ASME J. Appl. Mech."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1905","DOI":"10.4310\/CMS.2015.v13.n7.a11","article-title":"Global weak solutions to 1d compressible euler equations with radiation","volume":"13","author":"Blanc","year":"2015","journal-title":"Commun. Math. Sci."},{"key":"ref_5","first-page":"119","article-title":"Global existence of smooth solutions to the Euler-Boltzmann equations","volume":"43","author":"Pu","year":"2022","journal-title":"Chin. Ann. Math. Ser. A"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1166","DOI":"10.1016\/j.jmaa.2017.10.001","article-title":"Blow-up criteria for three-dimensional compressible radiation hydrodynamics equations with vacuum","volume":"458","author":"Li","year":"2018","journal-title":"J. Math. Anal. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1007\/s00021-005-0213-3","article-title":"Local existence and finite-time blow-up in multidimensional radiation hydrodynamics","volume":"9","author":"Zhong","year":"2007","journal-title":"J. Math. Fluid Mech."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2191","DOI":"10.1016\/j.jde.2012.06.014","article-title":"Global weak solutions to the non-relativistic radiation hydrodynamical equations with isothermal fluids","volume":"253","author":"Jiang","year":"2012","journal-title":"J. Differ. Equ."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1090\/S0033-569X-2011-01227-2","article-title":"Global weak solutions to the Euler-Boltzmann equations in radiation hydrodynamics","volume":"70","author":"Jiang","year":"2012","journal-title":"Q. Appl. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"809","DOI":"10.1088\/0951-7715\/23\/4\/003","article-title":"Formation of singularities of solutions to the three-dimensional Euler-Boltzmann equations in radiation hydrodynamics","volume":"23","author":"Jiang","year":"2010","journal-title":"Nonlinearity"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1016\/j.jmaa.2012.03.027","article-title":"Formation of singularities of solutions to the three-dimensional non-relativistic radiation hydrodynamic equations","volume":"395","author":"Jiang","year":"2012","journal-title":"J. Math. Anal. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3943","DOI":"10.1016\/j.jde.2014.03.007","article-title":"Formation of singularities in solutions to the compressible radiation hydrodynamics equations with vacuum","volume":"256","author":"Li","year":"2014","journal-title":"J. Differ. Equ."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1007\/s11401-021-0273-6","article-title":"On Blow-up of Regular Solutions to the Isentropic Euler and Euler-Boltzmann Equations with Vacuum","volume":"42","author":"Cao","year":"2021","journal-title":"Chin. Ann. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1142\/S0218202599000063","article-title":"Cauchy problem for a model system of radiating gas: Weak solution with a jump and classical solutions","volume":"9","author":"Kawashima","year":"1999","journal-title":"Math. Models Methods Appl. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1137\/S0036141097322169","article-title":"Shock wave for a model system of the radiation gas","volume":"30","author":"Kawashima","year":"1998","journal-title":"SIAM J. Math. Anal."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1007\/BF00280033","article-title":"Formation of singularities in solutions to nonlinear hyperbolic equations","volume":"86","author":"Sideris","year":"1984","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1007\/BF01210741","article-title":"Formation of singularities in three-dimensional compressible fluids","volume":"101","author":"Sideris","year":"1985","journal-title":"Commun. Math. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"795","DOI":"10.1081\/PDE-120020497","article-title":"Long time behavior of solution to the 3D compressible Euler equations with damping","volume":"28","author":"Sideris","year":"2003","journal-title":"Commun. Partial. Differ. Equ."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jde.2014.03.006","article-title":"Spreading of the free boundary of an ideal fluid in a vacuum","volume":"257","author":"Sideris","year":"2014","journal-title":"J. Differ. Equ."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"113445","DOI":"10.1016\/j.na.2023.113445","article-title":"Singularities in Finite Time of the Full Compressible Euler Equations in Rd","volume":"240","author":"Wu","year":"2024","journal-title":"Nonlinear Anal."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1007\/s00021-012-0116-z","article-title":"Irrotational Blowup of the Solution to Compressible Euler Equation","volume":"15","author":"Suzuki","year":"2013","journal-title":"J. Math. Fluid Mech."},{"key":"ref_22","first-page":"55","article-title":"Singularities of solutions to compressible Euler equations with vacuum","volume":"20","author":"Lei","year":"2012","journal-title":"Math. Res. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.na.2015.11.021","article-title":"Blowup for the 3D compressible Euler equations","volume":"133","author":"Zhu","year":"2016","journal-title":"Nonlinear Anal."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1016\/j.na.2017.04.007","article-title":"Blowup for irrotational C1 solutions of the compressible Euler equations in RN","volume":"158","author":"Yuen","year":"2017","journal-title":"Nonlinear Anal."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"041502","DOI":"10.1063\/1.5031120","article-title":"Blowup phenomenon for the initial-boundary value problem of the non-isentropic compressible Euler equations","volume":"59","author":"Cheung","year":"2018","journal-title":"J. Math. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1080\/00036811.2018.1506103","article-title":"Blowup for the compressible isothermal Euler equations with non-vacuum initial data","volume":"99","author":"Dong","year":"2020","journal-title":"Appl. Anal."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Liu, J., Qin, Z., and Yuen, M. (2024). Formation of Singularity for Isentropic Irrotational Compressible Euler Equations. Symmetry, 16.","DOI":"10.3390\/sym16040454"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1007\/BF00280740","article-title":"The Cauchy problem for quasi-linear symmetric hyperbolic systems","volume":"58","author":"Kato","year":"1975","journal-title":"Arch. Rational Mech. Anal."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Majda, A. (1984). Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer Science & Business Media.","DOI":"10.1007\/978-1-4612-1116-7"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/BF00250741","article-title":"Development of singularities in the motion of materials with fading memory","volume":"91","author":"Dafermos","year":"1986","journal-title":"ARMA"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1002\/cpa.3160270307","article-title":"Formation of singularities in one-dimensional nonlinear wave propagation","volume":"27","author":"John","year":"1974","journal-title":"Commun. Pure Appl. Math."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1002\/cpa.3160330304","article-title":"Formation of singularities for wave equations including the nonlinear vibrating string","volume":"33","author":"Klainerman","year":"1980","journal-title":"Commun. Pure Appl. Math."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1063\/1.1704154","article-title":"Development of singularities in solutions of nonlinear hyperbolic partial differential equations","volume":"5","author":"Lax","year":"1964","journal-title":"J. Math. Phys."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/6\/835\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:41:24Z","timestamp":1760031684000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/6\/835"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,27]]},"references-count":33,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2025,6]]}},"alternative-id":["sym17060835"],"URL":"https:\/\/doi.org\/10.3390\/sym17060835","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,5,27]]}}}