Asymptotic analysis of transmission problems with parameter-dependent Robin conditions
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866915611725004800 |
|---|---|
| author | Fukao, Takeshi |
| author_facet | Fukao, Takeshi |
| contents | We study a transmission problem of {N}eumann--{R}obin type involving a parameter $α$ and perform an asymptotic analysis with respect to $α$. The limits $α\to 0$ and $α\to +\infty$ correspond respectively to complete decoupling and full unification of the problem, and we obtain rates of convergence for both regimes. Biologically, the model describes two cells connected by a gap junction with permeability $α$: the case $α\to 0$ corresponds to a situation where the gap junction is closed, leaving only tight junctions between the cells so that no substance exchange occurs, while $α\to +\infty$ corresponds to a situation that can be interpreted as the cells forming a single structure. We also clarify the relationship between the asymptotic analysis with respect to the parameter $α$ and the asymptotics of the system in connection with the convergence of convex functionals known as {M}osco convergence. Finally, we consider time-dependent permeability and analyze the case where $α$ blows up in finite time. Under suitable regularity assumptions, we show that the solution can be extended beyond the blow-up time, remaining in the single structure regime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_07704 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic analysis of transmission problems with parameter-dependent Robin conditions Fukao, Takeshi Analysis of PDEs 34G25, 35K20, 35M13, 35Q92 We study a transmission problem of {N}eumann--{R}obin type involving a parameter $α$ and perform an asymptotic analysis with respect to $α$. The limits $α\to 0$ and $α\to +\infty$ correspond respectively to complete decoupling and full unification of the problem, and we obtain rates of convergence for both regimes. Biologically, the model describes two cells connected by a gap junction with permeability $α$: the case $α\to 0$ corresponds to a situation where the gap junction is closed, leaving only tight junctions between the cells so that no substance exchange occurs, while $α\to +\infty$ corresponds to a situation that can be interpreted as the cells forming a single structure. We also clarify the relationship between the asymptotic analysis with respect to the parameter $α$ and the asymptotics of the system in connection with the convergence of convex functionals known as {M}osco convergence. Finally, we consider time-dependent permeability and analyze the case where $α$ blows up in finite time. Under suitable regularity assumptions, we show that the solution can be extended beyond the blow-up time, remaining in the single structure regime. |
| title | Asymptotic analysis of transmission problems with parameter-dependent Robin conditions |
| topic | Analysis of PDEs 34G25, 35K20, 35M13, 35Q92 |
| url | https://arxiv.org/abs/2511.07704 |