{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T00:36:08Z","timestamp":1759970168681,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,6]],"date-time":"2025-01-06T00:00:00Z","timestamp":1736121600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100009110","name":"Natural Science Foundation of Xinjiang Uygur Autonomous Region","doi-asserted-by":"publisher","award":["2024D01C03"],"award-info":[{"award-number":["2024D01C03"]}],"id":[{"id":"10.13039\/100009110","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Chaplygin gas magnetohydrodynamics Euler equations with active terms are often used to study physical phenomena in the universe, which is of great significance for exploring unknown fields. This article mainly studies the limited behavior of Riemann solutions of Chaplygin gas magnetohydrodynamics Euler equations with active terms using methods such as the characteristic line method and the Lax\u2013Friedrichs method. In cases where only the magnetic field disappears, it was found, using the characteristic line method, that the solution converges to Chaplygin gas magnetohydrodynamics Euler equations with active terms. Additionally, we have identified the cause of the generation of \u03b4 shock waves. When pressure and magnetic induction disappear simultaneously, the reasons for the generation of \u03b4 shock waves and vacuum solutions are found. In the Discussion section, the Lax\u2013Friedrichs method was used for numerical experiments to simulate the occurrence of phenomena such as mass concentration and vacuum. The innovation of this article lies in the construction of the Riemann problem for Chaplygin gas MHD Euler equations with active terms, as well as the study of the scenario where the magnetic field and pressure gradually weaken and approach zero. Finally, numerical experiments are used to verify the theoretical results. Compared with previous work, this article not only focuses on theoretical derivation, but also applies numerical simulation, especially simulating the characteristic lines of physical planes. This achievement provides a powerful tool for studying more complex Riemann problems.<\/jats:p>","DOI":"10.3390\/sym17010077","type":"journal-article","created":{"date-parts":[[2025,1,6]],"date-time":"2025-01-06T10:40:52Z","timestamp":1736160052000},"page":"77","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Limits of Riemann Solutions for Chaplygin Gas Magnetohydrodynamics Euler Equations with Active Terms"],"prefix":"10.3390","volume":"17","author":[{"given":"Kangning","family":"Yu","sequence":"first","affiliation":[{"name":"Department of Mathematics, Changji University, Changji 831100, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wenxia","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541002, China"},{"name":"Center for Applied Mathematics of Guangxi (GUET), Guilin 541002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jibin","family":"Yang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Changji University, Changji 831100, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shuo","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Changji University, Changji 831100, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1275-2982","authenticated-orcid":false,"given":"Guodong","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Changji University, Changji 831100, China"},{"name":"School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541002, China"},{"name":"Center for Applied Mathematics of Guangxi (GUET), Guilin 541002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,6]]},"reference":[{"key":"ref_1","first-page":"159","article-title":"How could a rotating body such as the Sun become a magnet","volume":"87","author":"Larmor","year":"1919","journal-title":"Rep. Br. Assoc. Adv. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"768","DOI":"10.1093\/mnras\/94.8.768","article-title":"The Stability of Gaseous Stars","volume":"94","author":"Cowling","year":"1934","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"458","DOI":"10.1093\/mnras\/97.6.458","article-title":"The Non-uniform Rotation of the Sun and its Magnetic Fiel","volume":"97","author":"Ferraro","year":"1937","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_4","first-page":"1","article-title":"Theory of the laminar flow of an electrically conductive liquid in a homogeneous magnetic field","volume":"6","author":"Hartmann","year":"1937","journal-title":"Math.-Fys. Meddelelser"},{"key":"ref_5","first-page":"85","article-title":"On the cosmogony of the solar system","volume":"14","year":"1942","journal-title":"Stock. Obs. Ann."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1017\/S0305004100028139","article-title":"Steady motion of conducting fluids in pipes under transverse magneticfields","volume":"49","author":"Shercliff","year":"1953","journal-title":"Proc. Cambridge Philos. Soc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"644","DOI":"10.1017\/S0022112056000421","article-title":"The flow of conducting fluids in circular pipes under transversemagnetic fields","volume":"1","author":"Shercliff","year":"1956","journal-title":"J. Fluid Mech."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1002\/asna.2103010302","article-title":"Mean-field approach to spherical dynamo models","volume":"301","author":"Radler","year":"1980","journal-title":"Astronom. Nachr."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2009","DOI":"10.1016\/j.ijheatmasstransfer.2005.11.033","article-title":"An experimental study of thermally induced convection of molten gallium in magnetic fields","volume":"49","author":"Xu","year":"2006","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1016\/S0920-3796(02)00291-0","article-title":"Progress in coating development for fusion systems","volume":"61","author":"Smith","year":"2002","journal-title":"Fusion Eng. Des."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"634","DOI":"10.1016\/0920-3796(94)00150-6","article-title":"The effects of imperfect insulator coatings on MHD and heat transfer in rectangular ducts","volume":"27","author":"Ying","year":"1994","journal-title":"Fusion Eng. Des."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/s13226-016-0172-9","article-title":"Elementary wave interactions in magnetogasdynamics","volume":"47","author":"Liu","year":"2016","journal-title":"Indian J. Pure Appl. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"454","DOI":"10.1016\/j.jmaa.2012.07.064","article-title":"Riemann problem and wave interactions in magnetogasdynamics","volume":"397","author":"Liu","year":"2013","journal-title":"J. Math. Anal. Appl."},{"key":"ref_14","first-page":"76","article-title":"The Riemann problem for the pressureless Euler system with the Coulomb-like friction term","volume":"81","author":"Shen","year":"2016","journal-title":"IMA J. Appl. Math."},{"key":"ref_15","first-page":"654","article-title":"The Riemann problem for the transportation equations in gas dynamics","volume":"137","author":"Sheng","year":"1999","journal-title":"Mem. Amer. Math. Soc."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1007\/BF02101897","article-title":"Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics","volume":"177","author":"Weinan","year":"1996","journal-title":"Commun. Math. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1103\/RevModPhys.61.185","article-title":"Large-scale structure of the universe: Turbulence, intermittency, structures in a self-gravitating medium","volume":"61","author":"Sharin","year":"1989","journal-title":"Rev. Mod. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1002\/zamm.201500015","article-title":"The Riemann problem for the Chaplygin gas equations with a source term","volume":"96","author":"Shen","year":"2016","journal-title":"ZAMM-J. Appl. Math. Mech."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"326","DOI":"10.1007\/s00021-005-0162-x","article-title":"Solutions with concentration to the Riemann problem for the one-dimensional Chaplygin gas equations","volume":"7","author":"Brenier","year":"2005","journal-title":"J. Math. Fluid Mech."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2146","DOI":"10.1137\/110838091","article-title":"Two-dimensional Riemann problems for Chaplygin gas","volume":"44","author":"Chen","year":"2012","journal-title":"SIAM J. Math. Anal."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"588","DOI":"10.1016\/j.nonrwa.2017.11.013","article-title":"The Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations with a source term","volume":"41","author":"Guo","year":"2018","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"431","DOI":"10.3934\/cpaa.2010.9.431","article-title":"The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system","volume":"9","author":"Guo","year":"2010","journal-title":"Commun. Pure Appl. Anal."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"434","DOI":"10.1016\/j.jmaa.2013.02.026","article-title":"The Riemann problem for one dimensional generalized Chaplygin gas dynamics","volume":"403","author":"Wang","year":"2013","journal-title":"J. Math. Anal. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1408","DOI":"10.1002\/nag.2124","article-title":"Exact solution for granular flows","volume":"37","author":"Faccanoni","year":"2012","journal-title":"Int. J. Numer. Anal. Methods Geomech"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"925","DOI":"10.1137\/S0036141001399350","article-title":"Formation of \u03b4\u2013shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids","volume":"34","author":"Chen","year":"2003","journal-title":"SIAM J. Math. Anal."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1016\/S0893-9659(00)00187-7","article-title":"Note on the compressible Euler equations with zero temperature","volume":"14","author":"Li","year":"2001","journal-title":"Appl. Math. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1124","DOI":"10.1016\/j.aml.2011.01.038","article-title":"The limits of Riemann solutions to the isentropic magnetogasdynamics","volume":"24","author":"Shen","year":"2011","journal-title":"Appl. Math. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"127","DOI":"10.3934\/cpaa.2018008","article-title":"The Riemann problem and the limit solutions as magnetic field vanishes to magnetogasdynamics for generalized Chaplygin gas","volume":"17","author":"Chen","year":"2018","journal-title":"Commun. Pure Appl. Anal."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.nonrwa.2014.08.007","article-title":"Delta wave and vacuum state for generalized Chaplygin gas dynamics system as pressure vanishes","volume":"22","author":"Sheng","year":"2015","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"800","DOI":"10.1016\/j.jmaa.2013.12.025","article-title":"Delta-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations for modified Chaplygin gas","volume":"413","author":"Yang","year":"2014","journal-title":"J. Math. Anal. Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"3024","DOI":"10.1016\/j.jde.2010.09.004","article-title":"Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed Aw-Rascle model","volume":"249","author":"Shen","year":"2010","journal-title":"J. Differ. Equations"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/j.jmaa.2017.05.048","article-title":"The limit behavior of the Riemann solutions to the generalized Chaplygin gas equations with a source term","volume":"455","author":"Guo","year":"2017","journal-title":"J. Math. Anal. Appl."},{"key":"ref_33","first-page":"1","article-title":"Delta Shock Waves as Flux-Approximation Limit of Solutions to the Modified Chaplygin Gas Equations","volume":"168","author":"Liu","year":"2019","journal-title":"Acta Appl. Math."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s40840-021-01133-8","article-title":"Zero-exponent Limit to the Extended Chaplygin Gas Equations with Friction","volume":"44","author":"Zhang","year":"2021","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s13324-022-00692-8","article-title":"Delta shocks and vacuum states for the isentropic magnetogasdynamics equations for Chaplygin gas as pressure and magnetic field vanish","volume":"12","author":"Shao","year":"2022","journal-title":"Anal. Math. Phys."},{"key":"ref_36","first-page":"115513","article-title":"Delta-shocks and vacuums in Riemann solutions to the Umami Chaplygin Aw\u2013Rascle model","volume":"188","author":"Li","year":"2024","journal-title":"Chaos Solitons Fractals Interdiscip. J. Nonlinear Sci. Nonequilibrium Complex Phenom."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1515\/zna-2023-0181","article-title":"Delta-shock for the Chaplygin gas Euler equations with source terms","volume":"79","author":"Li","year":"2024","journal-title":"Z. F\u00fcr Naturforschung"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1186\/s13661-018-1009-8","article-title":"Riemann problem for a compressible perfect fluid with a constant external force for the Chaplygin gas","volume":"1","author":"Pang","year":"2018","journal-title":"Bound. Value Probl."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"295","DOI":"10.3934\/cpaa.2017014","article-title":"The vanishing pressure limits of Riemann solutions to the Chaplygin gas equations with a source term","volume":"16","author":"Guo","year":"2016","journal-title":"Commun. Pure Appl. Anal."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/1\/77\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:23:41Z","timestamp":1759919021000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/1\/77"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,6]]},"references-count":39,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["sym17010077"],"URL":"https:\/\/doi.org\/10.3390\/sym17010077","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,1,6]]}}}