A new formula for the Wasserstein distance between solutions to (nonlinear) continuity equations
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866915892791607296 |
|---|---|
| author | Carrillo, José A. Gwiazda, Piotr Skrzeczkowski, Jakub |
| author_facet | Carrillo, José A. Gwiazda, Piotr Skrzeczkowski, Jakub |
| contents | Given two continuity equations with density-dependent velocities, we provide a new formula for the Wasserstein distance between the solutions in terms of the difference of velocities evaluated at the same density. The formula is particularly attractive to deduce quantitative estimates and rates of convergence for singular limits. We illustrate it using several examples. For the porous medium equation with exponent $m$, we prove that solutions are Lipschitz continuous with respect to $m$, providing a quantitative version of the result of Bénilan and Crandall. This result can be extended to a general aggregation-diffusion equation. We also study the limit $m \to \infty$ (the so-called mesa problem or the incompressible limit) and we recover the rate of convergence $1/{\sqrt{m}}$. Last but not least, we improve the rate of nonlocal-to-local convergence for the quadratic porous medium equation from recently obtained $\sqrt{\varepsilon}$ to numerically conjectured $\varepsilon$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_25634 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A new formula for the Wasserstein distance between solutions to (nonlinear) continuity equations Carrillo, José A. Gwiazda, Piotr Skrzeczkowski, Jakub Analysis of PDEs 35A15, 35B30, 35B35, 35K65, 35B40, 35Q70 Given two continuity equations with density-dependent velocities, we provide a new formula for the Wasserstein distance between the solutions in terms of the difference of velocities evaluated at the same density. The formula is particularly attractive to deduce quantitative estimates and rates of convergence for singular limits. We illustrate it using several examples. For the porous medium equation with exponent $m$, we prove that solutions are Lipschitz continuous with respect to $m$, providing a quantitative version of the result of Bénilan and Crandall. This result can be extended to a general aggregation-diffusion equation. We also study the limit $m \to \infty$ (the so-called mesa problem or the incompressible limit) and we recover the rate of convergence $1/{\sqrt{m}}$. Last but not least, we improve the rate of nonlocal-to-local convergence for the quadratic porous medium equation from recently obtained $\sqrt{\varepsilon}$ to numerically conjectured $\varepsilon$. |
| title | A new formula for the Wasserstein distance between solutions to (nonlinear) continuity equations |
| topic | Analysis of PDEs 35A15, 35B30, 35B35, 35K65, 35B40, 35Q70 |
| url | https://arxiv.org/abs/2603.25634 |