Analysis of a Poisson-Nernst-Planck cross-diffusion system with steric effects
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910716771958784 |
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| author | Hirvonen, Peter Jüngel, Ansgar |
| author_facet | Hirvonen, Peter Jüngel, Ansgar |
| contents | A transient Poisson-Nernst-Planck system with steric effects is analyzed in a bounded domain with no-flux boundary conditions for the ion concentrations and mixed Dirichlet-Neumann boundary conditions for the electric potential. The steric repulsion of ions is modeled by a localized Lennard-Jones force, leading to cross-diffusion terms. The existence of global weak solutions, a weak--strong uniqueness property, and, in case of pure Neumann conditions, the exponential decay towards the thermal equilibrium state is proved. The main difficulties are the cross-diffusion terms and the different boundary conditions satisfied by the unknowns. These issues are overcome by exploiting the entropy structure of the equations and carefully taking into account the electric potential term. A numerical experiment illustrates the long-time behavior of the solutions when the potential satisfies mixed boundary conditions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_17399 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Analysis of a Poisson-Nernst-Planck cross-diffusion system with steric effects Hirvonen, Peter Jüngel, Ansgar Analysis of PDEs 35K51, 35A02, 35B40, 35Q92 A transient Poisson-Nernst-Planck system with steric effects is analyzed in a bounded domain with no-flux boundary conditions for the ion concentrations and mixed Dirichlet-Neumann boundary conditions for the electric potential. The steric repulsion of ions is modeled by a localized Lennard-Jones force, leading to cross-diffusion terms. The existence of global weak solutions, a weak--strong uniqueness property, and, in case of pure Neumann conditions, the exponential decay towards the thermal equilibrium state is proved. The main difficulties are the cross-diffusion terms and the different boundary conditions satisfied by the unknowns. These issues are overcome by exploiting the entropy structure of the equations and carefully taking into account the electric potential term. A numerical experiment illustrates the long-time behavior of the solutions when the potential satisfies mixed boundary conditions. |
| title | Analysis of a Poisson-Nernst-Planck cross-diffusion system with steric effects |
| topic | Analysis of PDEs 35K51, 35A02, 35B40, 35Q92 |
| url | https://arxiv.org/abs/2411.17399 |