Rate of convergence of the vanishing viscosity method for Hamilton-Jacobi equations with Neumann boundary conditions
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911378360500224 |
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| author | Goffi, Alessandro |
| author_facet | Goffi, Alessandro |
| contents | We study the quantitative small noise limit in the $L^\infty$ norm of certain time-dependent Hamilton-Jacobi equations equipped with Neumann boundary conditions, depending on the regularity of the data and the geometric properties of the domain. We first provide a $\mathcal{O}(\sqrt{\varepsilon})$ rate of convergence for Hamilton-Jacobi equations with locally Lipschitz Hamiltonians posed on convex domains of the Euclidean space. We then enhance this speed of convergence in the case of quadratic Hamiltonians proving one-side rates of order $\mathcal{O}(\varepsilon)$ and $\mathcal{O}(\varepsilon^β)$, $β\in(1/2,1)$. The results exploit recent $L^1$ contraction estimates for Fokker-Planck equations with bounded velocity fields on unbounded domains used to derive differential Harnack estimates for the corresponding Neumann heat flow. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_01973 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rate of convergence of the vanishing viscosity method for Hamilton-Jacobi equations with Neumann boundary conditions Goffi, Alessandro Analysis of PDEs We study the quantitative small noise limit in the $L^\infty$ norm of certain time-dependent Hamilton-Jacobi equations equipped with Neumann boundary conditions, depending on the regularity of the data and the geometric properties of the domain. We first provide a $\mathcal{O}(\sqrt{\varepsilon})$ rate of convergence for Hamilton-Jacobi equations with locally Lipschitz Hamiltonians posed on convex domains of the Euclidean space. We then enhance this speed of convergence in the case of quadratic Hamiltonians proving one-side rates of order $\mathcal{O}(\varepsilon)$ and $\mathcal{O}(\varepsilon^β)$, $β\in(1/2,1)$. The results exploit recent $L^1$ contraction estimates for Fokker-Planck equations with bounded velocity fields on unbounded domains used to derive differential Harnack estimates for the corresponding Neumann heat flow. |
| title | Rate of convergence of the vanishing viscosity method for Hamilton-Jacobi equations with Neumann boundary conditions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2509.01973 |