On the vanishing viscosity limit of Hamilton-Jacobi equations with nearly optimal discount

Zibo Wang; Jianlu Zhang

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arXiv · 2509.17402

On the vanishing viscosity limit of Hamilton-Jacobi equations with nearly optimal discount

Zibo Wang·Jianlu Zhang

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Abstract

In this paper, we establish the convergence of solutions to the viscous Hamilton-Jacobi equation (with a Tonelli Hamiltonian): \[ \lambda u +H(x, du)=\varepsilon(\lambda)\Delta u,\quad \lambda>0 \] as $\lambda\rightarrow 0_+$, once the modulus $\varepsilon(\lambda)$ satisfies $\varlimsup_{\lambda\rightarrow 0_+}\varepsilon(\lambda)/\lambda=0$. Such an exponent of $\varepsilon(\lambda)$ is nearly optimal in the convergence.

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Zibo Wang, Jianlu Zhang. 2025-09-22. On the vanishing viscosity limit of Hamilton-Jacobi equations with nearly optimal discount. https://arxiv.org/abs/2509.17402

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