Convergence rates for the vanishing viscosity approximation of fully nonlinear, non-convex, second-order Hamilton-Jacobi equations

2025/09/15 by Alekos Cecchin, Cecchin, Alekos, Alessandro Goffi +1
Computer Science · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.2509.12144

openalex publication_date 2025/09/15 · openalex created_date 2025/10/12 · openalex updated_date 2026/07/28

Abstract

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 ∂_t u+H(x,t,Du,D²u)=0 in the whole space. We approximate the PDE with a fully nonlinear, possibly degenerate, elliptic operator ε F(x,t,D²u). Assuming that u∈ C^α_x, u₀∈ C^η, H∈ C^β_x and having power growth γ in the gradient entry, we establish a convergence rate of order ε^min\\fracη2,(β+γ(α-1))/(β+γ(α-1)+2-α)\. 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.

Citations

• Optimal rate of convergence in the vanishing viscosity for uniformly convex Hamilton-Jacobi equations
• Convergence rates for the vanishing viscosity approximation of Hamilton-Jacobi equations: the convex case
• Transport equations and flows with one-sided Lipschitz velocity fields
• L¹-Stability and error estimates for approximate Hamilton-Jacobi solutions
• Two approximations of solutions of Hamilton-Jacobi equations

Cited by

• On the rate of convergence in superquadratic Hamilton--Jacobi equations with state constraints

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