The Aronson-Bénilan estimate for a Lagrangian particle discretization of the Porous Medium Equation
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912884899971072 |
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| author | Di Francesco, Marco Matthes, Daniel |
| author_facet | Di Francesco, Marco Matthes, Daniel |
| contents | We consider a nearest neighbor, Lagrangian particle discretization of the one dimensional porous medium equation. We prove that the particle model satisfies a discrete analog of the celebrated Aronson-Bénilan estimate, which we use to prove a growth estimate for the evolution of the support and an $L^\infty$ decay estimate which are both known to hold in the continuum. These estimates are uniform with respect to the number of particles. We also prove convergence of the scheme towards the solution to the porous medium equation in the full generality of $L^1$ initial data. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_06835 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Aronson-Bénilan estimate for a Lagrangian particle discretization of the Porous Medium Equation Di Francesco, Marco Matthes, Daniel Analysis of PDEs 35K55, 35K65, 35Q70, 35A35 We consider a nearest neighbor, Lagrangian particle discretization of the one dimensional porous medium equation. We prove that the particle model satisfies a discrete analog of the celebrated Aronson-Bénilan estimate, which we use to prove a growth estimate for the evolution of the support and an $L^\infty$ decay estimate which are both known to hold in the continuum. These estimates are uniform with respect to the number of particles. We also prove convergence of the scheme towards the solution to the porous medium equation in the full generality of $L^1$ initial data. |
| title | The Aronson-Bénilan estimate for a Lagrangian particle discretization of the Porous Medium Equation |
| topic | Analysis of PDEs 35K55, 35K65, 35Q70, 35A35 |
| url | https://arxiv.org/abs/2602.06835 |