Convergence rates for the vanishing viscosity approximation of fully nonlinear, non-convex, second-order Hamilton-Jacobi equations
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arXiv
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| Format: | Preprint |
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2025
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| author | Cecchin, Alekos Goffi, Alessandro |
| author_facet | Cecchin, Alekos Goffi, Alessandro |
| contents | We obtain new quantitative estimates of the vanishing viscosity approximation for time-dependent, degenerate, Hamilton-Jacobi equations that are neither concave nor convex in the gradient and Hessian entries of the form $\partial_t u+H(x,t,Du,D^2u)=0$ in the whole space. We approximate the PDE with a fully nonlinear, possibly degenerate, elliptic operator $\varepsilon F(x,t,D^2u)$. Assuming that $u\in C^α_x$, $u_0\in C^η$, $H\in C^β_x$ and having power growth $γ$ in the gradient entry, we establish a convergence rate of order $\varepsilon^{\min\left\{\fracη{2},\frac{β+γ(α-1)}{β+γ(α-1)+2-α}\right\}}$. Our novel approach exploits the regularizing properties of sup/inf-convolutions for viscosity solutions and the comparison principle. We also obtain explicit constants and do not assume differentiability properties neither on solutions nor on $H$. The same method provides new convergence rates for the vanishing viscosity approximation of the stationary counterpart of the equation and for transport equations with Hölder coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_12144 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence rates for the vanishing viscosity approximation of fully nonlinear, non-convex, second-order Hamilton-Jacobi equations Cecchin, Alekos Goffi, Alessandro Analysis of PDEs We obtain new quantitative estimates of the vanishing viscosity approximation for time-dependent, degenerate, Hamilton-Jacobi equations that are neither concave nor convex in the gradient and Hessian entries of the form $\partial_t u+H(x,t,Du,D^2u)=0$ in the whole space. We approximate the PDE with a fully nonlinear, possibly degenerate, elliptic operator $\varepsilon F(x,t,D^2u)$. Assuming that $u\in C^α_x$, $u_0\in C^η$, $H\in C^β_x$ and having power growth $γ$ in the gradient entry, we establish a convergence rate of order $\varepsilon^{\min\left\{\fracη{2},\frac{β+γ(α-1)}{β+γ(α-1)+2-α}\right\}}$. Our novel approach exploits the regularizing properties of sup/inf-convolutions for viscosity solutions and the comparison principle. We also obtain explicit constants and do not assume differentiability properties neither on solutions nor on $H$. The same method provides new convergence rates for the vanishing viscosity approximation of the stationary counterpart of the equation and for transport equations with Hölder coefficients. |
| title | Convergence rates for the vanishing viscosity approximation of fully nonlinear, non-convex, second-order Hamilton-Jacobi equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2509.12144 |