Monge-Kantorovich distance for PDEs: the coupling method
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2019
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916916426178560 |
|---|---|
| author | Fournier, Nicolas Perthame, Benoît |
| author_facet | Fournier, Nicolas Perthame, Benoît |
| contents | We informally review a few PDEs for which the Monge-Kantorovich distance between pairs of solutions, possibly with some judicious cost function, decays: heat equation, Fokker-Planck equation, heat equation with varying coefficients, fractional heat equation with varying coefficients, homogeneous Boltzmann equation for Maxwell molecules, and some nonlinear integro-differential equations arising in neurosciences. We always use the same method, that consists in building a coupling between two solutions. This amounts to solve a well-chosen PDE posed on the Euclidian square of the physical space, i.e. doubling the variables. Finally, although the above method fails, we recall a simple idea to treat the case of the porous media equation. We also introduce another method based on the dual Monge-Kantorovich problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1903_11349 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Monge-Kantorovich distance for PDEs: the coupling method Fournier, Nicolas Perthame, Benoît Analysis of PDEs Probability We informally review a few PDEs for which the Monge-Kantorovich distance between pairs of solutions, possibly with some judicious cost function, decays: heat equation, Fokker-Planck equation, heat equation with varying coefficients, fractional heat equation with varying coefficients, homogeneous Boltzmann equation for Maxwell molecules, and some nonlinear integro-differential equations arising in neurosciences. We always use the same method, that consists in building a coupling between two solutions. This amounts to solve a well-chosen PDE posed on the Euclidian square of the physical space, i.e. doubling the variables. Finally, although the above method fails, we recall a simple idea to treat the case of the porous media equation. We also introduce another method based on the dual Monge-Kantorovich problem. |
| title | Monge-Kantorovich distance for PDEs: the coupling method |
| topic | Analysis of PDEs Probability |
| url | https://arxiv.org/abs/1903.11349 |