{"title":"激波对应的边界效应双曲-椭圆耦合系统平面平稳解的收敛速率","authors":"Shanming Ji, Minyi Zhang, Changjiang Zhu","doi":"10.1016/j.jde.2025.113492","DOIUrl":null,"url":null,"abstract":"<div><div>In this paper, we study the asymptotic behavior of solutions to an initial-boundary value problem for a hyperbolic-elliptic coupled system of the radiating gas on half space with the conditions <span><math><mi>u</mi><mo>(</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mo>−</mo></mrow></msub></math></span> and <span><math><mi>u</mi><mo>(</mo><mo>∞</mo><mo>,</mo><mi>y</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mo>+</mo></mrow></msub><mo><</mo><msub><mrow><mi>u</mi></mrow><mrow><mo>−</mo></mrow></msub></math></span>, where the corresponding Cauchy problem admits the shock wave as an asymptotic profile. In the case of <span><math><msub><mrow><mi>u</mi></mrow><mrow><mo>+</mo></mrow></msub><mo><</mo><msub><mrow><mi>u</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>≤</mo><mn>0</mn></math></span>, we prove that the solution to the problem converges to the corresponding planar stationary solution as time tends to infinity by assuming that the initial perturbation is small. Furthermore, we obtain the convergence rate by applying the time and space weighted energy method. The results include one-dimensional and two-dimensional cases.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"442 ","pages":"Article 113492"},"PeriodicalIF":2.4000,"publicationDate":"2025-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Convergence rates toward the planar stationary solution for a hyperbolic-elliptic coupled system with boundary effect corresponding to shock wave\",\"authors\":\"Shanming Ji, Minyi Zhang, Changjiang Zhu\",\"doi\":\"10.1016/j.jde.2025.113492\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>In this paper, we study the asymptotic behavior of solutions to an initial-boundary value problem for a hyperbolic-elliptic coupled system of the radiating gas on half space with the conditions <span><math><mi>u</mi><mo>(</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mo>−</mo></mrow></msub></math></span> and <span><math><mi>u</mi><mo>(</mo><mo>∞</mo><mo>,</mo><mi>y</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mo>+</mo></mrow></msub><mo><</mo><msub><mrow><mi>u</mi></mrow><mrow><mo>−</mo></mrow></msub></math></span>, where the corresponding Cauchy problem admits the shock wave as an asymptotic profile. In the case of <span><math><msub><mrow><mi>u</mi></mrow><mrow><mo>+</mo></mrow></msub><mo><</mo><msub><mrow><mi>u</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>≤</mo><mn>0</mn></math></span>, we prove that the solution to the problem converges to the corresponding planar stationary solution as time tends to infinity by assuming that the initial perturbation is small. Furthermore, we obtain the convergence rate by applying the time and space weighted energy method. The results include one-dimensional and two-dimensional cases.</div></div>\",\"PeriodicalId\":15623,\"journal\":{\"name\":\"Journal of Differential Equations\",\"volume\":\"442 \",\"pages\":\"Article 113492\"},\"PeriodicalIF\":2.4000,\"publicationDate\":\"2025-06-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Differential Equations\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022039625005194\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022039625005194","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Convergence rates toward the planar stationary solution for a hyperbolic-elliptic coupled system with boundary effect corresponding to shock wave
In this paper, we study the asymptotic behavior of solutions to an initial-boundary value problem for a hyperbolic-elliptic coupled system of the radiating gas on half space with the conditions and , where the corresponding Cauchy problem admits the shock wave as an asymptotic profile. In the case of , we prove that the solution to the problem converges to the corresponding planar stationary solution as time tends to infinity by assuming that the initial perturbation is small. Furthermore, we obtain the convergence rate by applying the time and space weighted energy method. The results include one-dimensional and two-dimensional cases.
期刊介绍:
The Journal of Differential Equations is concerned with the theory and the application of differential equations. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools.
Research Areas Include:
• Mathematical control theory
• Ordinary differential equations
• Partial differential equations
• Stochastic differential equations
• Topological dynamics
• Related topics