{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:55:29Z","timestamp":1760237729803,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,7,20]],"date-time":"2022-07-20T00:00:00Z","timestamp":1658275200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Guizhou Provincial Science and Technology Projects","award":["ZK[2022]535","(2016)4006","[2018]012","No. APVV-18-0308","No. 1\/0358\/20","No. 2\/0127\/20"],"award-info":[{"award-number":["ZK[2022]535","(2016)4006","[2018]012","No. APVV-18-0308","No. 1\/0358\/20","No. 2\/0127\/20"]}]},{"name":"Training Object of High Level and Innovative Talents of Guizhou Province","award":["ZK[2022]535","(2016)4006","[2018]012","No. APVV-18-0308","No. 1\/0358\/20","No. 2\/0127\/20"],"award-info":[{"award-number":["ZK[2022]535","(2016)4006","[2018]012","No. APVV-18-0308","No. 1\/0358\/20","No. 2\/0127\/20"]}]},{"name":"Major Research Project of Innovative Group in Guizhou Education Department","award":["ZK[2022]535","(2016)4006","[2018]012","No. APVV-18-0308","No. 1\/0358\/20","No. 2\/0127\/20"],"award-info":[{"award-number":["ZK[2022]535","(2016)4006","[2018]012","No. APVV-18-0308","No. 1\/0358\/20","No. 2\/0127\/20"]}]},{"name":"the Slovak Research and Development Agency","award":["ZK[2022]535","(2016)4006","[2018]012","No. APVV-18-0308","No. 1\/0358\/20","No. 2\/0127\/20"],"award-info":[{"award-number":["ZK[2022]535","(2016)4006","[2018]012","No. APVV-18-0308","No. 1\/0358\/20","No. 2\/0127\/20"]}]},{"name":"the Slovak Grant Agency VEGA","award":["ZK[2022]535","(2016)4006","[2018]012","No. APVV-18-0308","No. 1\/0358\/20","No. 2\/0127\/20"],"award-info":[{"award-number":["ZK[2022]535","(2016)4006","[2018]012","No. APVV-18-0308","No. 1\/0358\/20","No. 2\/0127\/20"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, based on the Euler equation and mass conservation equation in spherical coordinates, the ratio of the stratospheric average width to the planetary radius and the ratio of the vertical velocity to the horizontal velocity are selected as parameters under appropriate boundary conditions. We establish the approximate system using these two small parameters. In addition, we consider the time dependence of the system and establish the governing equations describing the atmospheric flow. By introducing a flow function to code the system, a nonlinear vorticity equation describing the planetary flow in the stratosphere is obtained. The governing equations describing the atmospheric flow are transformed into a second-order homogeneous linear ordinary differential equation and a Legendre\u2019s differential equation by applying the method of separating variables based on the concepts of spherical harmonic functions and weak solutions. The Gronwall inequality and the Cauchy\u2013Schwartz inequality are applied to priori estimates for the vorticity equation describing the stratospheric planetary flow under the appropriate initial and boundary conditions. The existence and non-uniqueness of weak solutions to the vorticity equation are obtained by using the functional analysis technique.<\/jats:p>","DOI":"10.3390\/axioms11070347","type":"journal-article","created":{"date-parts":[[2022,7,20]],"date-time":"2022-07-20T11:22:24Z","timestamp":1658316144000},"page":"347","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Existence of Weak Solutions for the Vorticity Equation Related to the Stratosphere in a Rotating Spherical Coordinate System"],"prefix":"10.3390","volume":"11","author":[{"given":"Wenlin","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Liupanshui Normal University, Liupanshui 553004, China"},{"name":"Department of Mathematics, Guizhou University, Guiyang 550025, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7385-6737","authenticated-orcid":false,"given":"Michal","family":"Fe\u010dkan","sequence":"additional","affiliation":[{"name":"Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynsk\u00e1 dolina, 842 48 Bratislava, Slovakia"},{"name":"Mathematical Institute, Slovak Academy of Sciences, \u0160tef\u00e1nikova 49, 814 73 Bratislava, Slovakia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jinrong","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Guizhou University, Guiyang 550025, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"751","DOI":"10.1016\/j.jde.2021.03.019","article-title":"On the modelling of large-scale atmospheric flow","volume":"285","author":"Constantin","year":"2021","journal-title":"J. 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