Remarks on the rate of convergence of the vanishing viscosity process of Hamilton-Jacobi equations
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915073175322624 |
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| author | Goffi, Alessandro |
| author_facet | Goffi, Alessandro |
| contents | We establish a linear $L^p$ rate of convergence, $1<p<\infty$, with respect to the viscosity $\varepsilon$ for the vanishing viscosity process of semiconcave solutions of Hamilton-Jacobi equations by regularizing the PDE with the half-Laplacian $-\varepsilon(-Δ)^{1/2}$. Our result reveals a nonlocal phenomenon, since it improves the known estimates obtained via the classical second order vanishing viscosity regularization $\varepsilonΔu$. It also highlights a faster rate of convergence than the available $\mathcal{O}(\varepsilon|\log\varepsilon|)$ rate in sup-norm obtained by the doubling of variable technique for this nonlocal approximation. The result is based on integral methods and does not use the maximum principle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15651 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Remarks on the rate of convergence of the vanishing viscosity process of Hamilton-Jacobi equations Goffi, Alessandro Analysis of PDEs We establish a linear $L^p$ rate of convergence, $1<p<\infty$, with respect to the viscosity $\varepsilon$ for the vanishing viscosity process of semiconcave solutions of Hamilton-Jacobi equations by regularizing the PDE with the half-Laplacian $-\varepsilon(-Δ)^{1/2}$. Our result reveals a nonlocal phenomenon, since it improves the known estimates obtained via the classical second order vanishing viscosity regularization $\varepsilonΔu$. It also highlights a faster rate of convergence than the available $\mathcal{O}(\varepsilon|\log\varepsilon|)$ rate in sup-norm obtained by the doubling of variable technique for this nonlocal approximation. The result is based on integral methods and does not use the maximum principle. |
| title | Remarks on the rate of convergence of the vanishing viscosity process of Hamilton-Jacobi equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.15651 |