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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2404.02623 |
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| _version_ | 1866916191152373760 |
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| author | Munoz, Sebastian |
| author_facet | Munoz, Sebastian |
| contents | We study the intermediate asymptotic behavior of solutions to the first-order mean field games system with a local coupling, when the initial density is a compactly supported function on the real line, and the coupling is of power type. Addressing a question that was left open in arXiv:2308.00314, we prove that the solutions converge to the self-similar profile. We proceed by analyzing a continuous rescaling of the solution, and identifying an appropriate Lyapunov functional. We identify a critical value for the parameter of the coupling, which determines the qualitative behavior of the functional, and the well-posedness of the infinite horizon system. Accordingly, we also establish, in the subcritical and critical cases, a second convergence result which characterizes the behavior of the full solution as the time horizon approaches infinity. We also prove the corresponding results for the mean field planning problem. A large part of our analysis and methodology apply just as well to arbitrary dimensions. As such, this work is a major step towards settling these questions in the higher-dimensional setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_02623 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Self-similar intermediate asymptotics for first-order mean field games Munoz, Sebastian Analysis of PDEs Optimization and Control 35Q89 (Primary) 35B40, 35R35, 35J70 (Secondary) We study the intermediate asymptotic behavior of solutions to the first-order mean field games system with a local coupling, when the initial density is a compactly supported function on the real line, and the coupling is of power type. Addressing a question that was left open in arXiv:2308.00314, we prove that the solutions converge to the self-similar profile. We proceed by analyzing a continuous rescaling of the solution, and identifying an appropriate Lyapunov functional. We identify a critical value for the parameter of the coupling, which determines the qualitative behavior of the functional, and the well-posedness of the infinite horizon system. Accordingly, we also establish, in the subcritical and critical cases, a second convergence result which characterizes the behavior of the full solution as the time horizon approaches infinity. We also prove the corresponding results for the mean field planning problem. A large part of our analysis and methodology apply just as well to arbitrary dimensions. As such, this work is a major step towards settling these questions in the higher-dimensional setting. |
| title | Self-similar intermediate asymptotics for first-order mean field games |
| topic | Analysis of PDEs Optimization and Control 35Q89 (Primary) 35B40, 35R35, 35J70 (Secondary) |
| url | https://arxiv.org/abs/2404.02623 |