The vanishing viscosity process for an eikonal equation in the radially symmetric setting

2025/08/17 by Meng, Fanchen
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.13230

Abstract

We study the vanishing viscosity method for the eikonal equation |Du|=V in B(0,1) with homogeneous Dirichlet boundary value condition. By assuming V is radially symmetric and restricting attention to radially symmetric solutions, we construct explicit formulas for both the viscous solution u^ε and the limiting solution u. We prove u^ε→ u as ε→ 0⁺ qualitatively and quantitatively derive an ε|log ε| type local convergence rate. Finally, we discuss the uniqueness of viscosity solutions for the eikonal equation and give some examples.

Citations

• Convergence rates for the vanishing viscosity approximation of Hamilton-Jacobi equations: the convex case
• Optimal rate of convergence in the vanishing viscosity for quadratic Hamilton-Jacobi equations
• On the regularity of stochastic effective Hamiltonian
• A Stochastic Analog of Aubry-Mather Theory

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