Long time dynamics of the Nernst-Planck-Darcy System on $\mathbb{R}^3$
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914234005192704 |
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| author | Abdo, Elie Germany, Joe Hamdan, Mohammad Khalil Kontar, Kifah |
| author_facet | Abdo, Elie Germany, Joe Hamdan, Mohammad Khalil Kontar, Kifah |
| contents | We study ionic electrodiffusion modeled by the Nernst--Planck equations describing the evolution of $N$ ionic species in a three-dimensional incompressible fluid flowing through a porous medium. We address the long-time dynamics of the resulting system in the three-dimensional whole space $\mathbb{R}^3$. We prove that the $k$-th spatial derivatives of each ionic concentration decays to zero in $L^2$ with a sharp rate of order $t^{-\frac{3}{4}-\frac{k}{2}}$. Moreover, we investigate the behavior of the relative entropy associated with the model and show that it blows up in time with a sharp growth rate of order $\log t$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_02208 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Long time dynamics of the Nernst-Planck-Darcy System on $\mathbb{R}^3$ Abdo, Elie Germany, Joe Hamdan, Mohammad Khalil Kontar, Kifah Analysis of PDEs 35K57, 35B40, 35Q92 We study ionic electrodiffusion modeled by the Nernst--Planck equations describing the evolution of $N$ ionic species in a three-dimensional incompressible fluid flowing through a porous medium. We address the long-time dynamics of the resulting system in the three-dimensional whole space $\mathbb{R}^3$. We prove that the $k$-th spatial derivatives of each ionic concentration decays to zero in $L^2$ with a sharp rate of order $t^{-\frac{3}{4}-\frac{k}{2}}$. Moreover, we investigate the behavior of the relative entropy associated with the model and show that it blows up in time with a sharp growth rate of order $\log t$. |
| title | Long time dynamics of the Nernst-Planck-Darcy System on $\mathbb{R}^3$ |
| topic | Analysis of PDEs 35K57, 35B40, 35Q92 |
| url | https://arxiv.org/abs/2601.02208 |