David M. Ambrose, Megan Griffin-Pickering, Alpár R. Mészáros
{"title":"具有不可分局部哈密顿量的运动型平均场对策","authors":"David M. Ambrose, Megan Griffin-Pickering, Alpár R. Mészáros","doi":"10.1112/jlms.70202","DOIUrl":null,"url":null,"abstract":"<p>We prove well-posedness of a class of <i>kinetic-type</i> mean field games (MFGs), which typically arise when agents control their acceleration. Such systems include independent variables representing the spatial position as well as velocity. We consider non-separable Hamiltonians without any structural conditions, which depend locally on the density variable. Our analysis is based on two main ingredients: an energy method for the forward–backward system in Sobolev spaces, on the one hand, and on a suitable <i>vector field method</i> to control derivatives with respect to the velocity variable, on the other hand. The careful combination of these two techniques reveals interesting phenomena applicable for MFGs involving general classes of drift-diffusion operators and non-linearities. While many prior existence theories for general MFGs systems take the final datum function to be smoothing, we can allow this function to be non-smoothing, that is, also depending locally on the final measure. Our well-posedness results hold under an appropriate smallness condition, assumed jointly on the data.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 6","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Kinetic-type mean field games with non-separable local Hamiltonians\",\"authors\":\"David M. Ambrose, Megan Griffin-Pickering, Alpár R. Mészáros\",\"doi\":\"10.1112/jlms.70202\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We prove well-posedness of a class of <i>kinetic-type</i> mean field games (MFGs), which typically arise when agents control their acceleration. Such systems include independent variables representing the spatial position as well as velocity. We consider non-separable Hamiltonians without any structural conditions, which depend locally on the density variable. Our analysis is based on two main ingredients: an energy method for the forward–backward system in Sobolev spaces, on the one hand, and on a suitable <i>vector field method</i> to control derivatives with respect to the velocity variable, on the other hand. The careful combination of these two techniques reveals interesting phenomena applicable for MFGs involving general classes of drift-diffusion operators and non-linearities. While many prior existence theories for general MFGs systems take the final datum function to be smoothing, we can allow this function to be non-smoothing, that is, also depending locally on the final measure. Our well-posedness results hold under an appropriate smallness condition, assumed jointly on the data.</p>\",\"PeriodicalId\":49989,\"journal\":{\"name\":\"Journal of the London Mathematical Society-Second Series\",\"volume\":\"111 6\",\"pages\":\"\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2025-06-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the London Mathematical Society-Second Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.70202\",\"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 the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.70202","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Kinetic-type mean field games with non-separable local Hamiltonians
We prove well-posedness of a class of kinetic-type mean field games (MFGs), which typically arise when agents control their acceleration. Such systems include independent variables representing the spatial position as well as velocity. We consider non-separable Hamiltonians without any structural conditions, which depend locally on the density variable. Our analysis is based on two main ingredients: an energy method for the forward–backward system in Sobolev spaces, on the one hand, and on a suitable vector field method to control derivatives with respect to the velocity variable, on the other hand. The careful combination of these two techniques reveals interesting phenomena applicable for MFGs involving general classes of drift-diffusion operators and non-linearities. While many prior existence theories for general MFGs systems take the final datum function to be smoothing, we can allow this function to be non-smoothing, that is, also depending locally on the final measure. Our well-posedness results hold under an appropriate smallness condition, assumed jointly on the data.
期刊介绍:
The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.