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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.02108 |
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| _version_ | 1866917558258499584 |
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| author | Xiao, Wang Feng, Lingyu Liu, Kai Zhao, Meng |
| author_facet | Xiao, Wang Feng, Lingyu Liu, Kai Zhao, Meng |
| contents | Hele-Shaw problems are prototypes to study the interface dynamics. Linear theory suggests the existence of self-similar patterns in a Hele-Shaw flow. That is, with a specific injection flux the interface shape remains unchanged while its size increases. In this paper, we explore the existence of self-similar patterns in the nonlinear regime and develop a rigorous nonlinear theory characterizing their fundamental features. Using a boundary integral formulation, we pose the question of self-similarity as a generalized nonlinear eigenvalue problem, involving two nonlinear integral operators. The flux constant $C$ is the eigenvalue and the corresponding self-similar pattern $\mathbf{x}$ is the eigenvector. We develop a quasi-Newton method to solve the problem and show the existence of nonlinear shapes with $k$-fold dominated symmetries. The influence of initial guesses on the self-similar patterns is investigated. We are able to obtain a desired self-similar shape once the initial guess is properly chosen. Our results go beyond the predictions of linear theory and establish a bridge between the linear theory and simulations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_02108 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An eigenvalue problem for self-similar patterns in Hele-Shaw flows Xiao, Wang Feng, Lingyu Liu, Kai Zhao, Meng Analysis of PDEs Hele-Shaw problems are prototypes to study the interface dynamics. Linear theory suggests the existence of self-similar patterns in a Hele-Shaw flow. That is, with a specific injection flux the interface shape remains unchanged while its size increases. In this paper, we explore the existence of self-similar patterns in the nonlinear regime and develop a rigorous nonlinear theory characterizing their fundamental features. Using a boundary integral formulation, we pose the question of self-similarity as a generalized nonlinear eigenvalue problem, involving two nonlinear integral operators. The flux constant $C$ is the eigenvalue and the corresponding self-similar pattern $\mathbf{x}$ is the eigenvector. We develop a quasi-Newton method to solve the problem and show the existence of nonlinear shapes with $k$-fold dominated symmetries. The influence of initial guesses on the self-similar patterns is investigated. We are able to obtain a desired self-similar shape once the initial guess is properly chosen. Our results go beyond the predictions of linear theory and establish a bridge between the linear theory and simulations. |
| title | An eigenvalue problem for self-similar patterns in Hele-Shaw flows |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2401.02108 |