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Rate of convergence of the vanishing viscosity method for Hamilton-Jacobi equations with Neumann boundary conditions

Alessandro Goffi

TL;DR

This work analyzes the rate at which the vanishing viscosity approximation converges to the Hamilton-Jacobi equation under Neumann boundary conditions on convex (possibly unbounded) domains. It combines duality with an adjoint Fokker-Planck problem and recent $L^1$ contraction results to obtain quantitative rates, first showing a robust $O\left(\sqrt{\\varepsilon T}\right)$ convergence for Lipschitz Hamiltonians, then proving improved one-sided rates $-C\\varepsilon^\\beta\le u_\\varepsilon-u \le C'\\varepsilon$ for quadratic Hamiltonians under semi-superharmonicity. The paper also establishes one-sided second-derivative bounds for solutions, independent of the viscosity, using integral methods and adjoint mass conservation, thereby enriching the theory of HJ equations with Neumann data in unbounded and convex geometries. These results have implications for numerical analysis and mean-field control problems where precise convergence rates of viscous approximations are essential.

Abstract

We study the quantitative small noise limit in the $L^\infty$ norm of certain time-dependent Hamilton-Jacobi equations equipped with Neumann boundary conditions, depending on the regularity of the data and the geometric properties of the domain. We first provide a $\mathcal{O}(\sqrt{\varepsilon})$ rate of convergence for Hamilton-Jacobi equations with locally Lipschitz Hamiltonians posed on convex domains of the Euclidean space. We then enhance this speed of convergence in the case of quadratic Hamiltonians proving one-side rates of order $\mathcal{O}(\varepsilon)$ and $\mathcal{O}(\varepsilon^β)$, $β\in(1/2,1)$. The results exploit recent $L^1$ contraction estimates for Fokker-Planck equations with bounded velocity fields on unbounded domains used to derive differential Harnack estimates for the corresponding Neumann heat flow.

Rate of convergence of the vanishing viscosity method for Hamilton-Jacobi equations with Neumann boundary conditions

TL;DR

This work analyzes the rate at which the vanishing viscosity approximation converges to the Hamilton-Jacobi equation under Neumann boundary conditions on convex (possibly unbounded) domains. It combines duality with an adjoint Fokker-Planck problem and recent contraction results to obtain quantitative rates, first showing a robust convergence for Lipschitz Hamiltonians, then proving improved one-sided rates for quadratic Hamiltonians under semi-superharmonicity. The paper also establishes one-sided second-derivative bounds for solutions, independent of the viscosity, using integral methods and adjoint mass conservation, thereby enriching the theory of HJ equations with Neumann data in unbounded and convex geometries. These results have implications for numerical analysis and mean-field control problems where precise convergence rates of viscous approximations are essential.

Abstract

We study the quantitative small noise limit in the norm of certain time-dependent Hamilton-Jacobi equations equipped with Neumann boundary conditions, depending on the regularity of the data and the geometric properties of the domain. We first provide a rate of convergence for Hamilton-Jacobi equations with locally Lipschitz Hamiltonians posed on convex domains of the Euclidean space. We then enhance this speed of convergence in the case of quadratic Hamiltonians proving one-side rates of order and , . The results exploit recent contraction estimates for Fokker-Planck equations with bounded velocity fields on unbounded domains used to derive differential Harnack estimates for the corresponding Neumann heat flow.

Paper Structure

This paper contains 4 sections, 5 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Preliminary results on Fokker-Planck equations with Neumann boundary conditions
  3. Rate of convergence estimates
  4. One-side second derivative bounds for solutions of Hamilton-Jacobi equations

Key Result

Theorem 2.1

[Theorem 2.4 in GoffiTralli] Assume that $\Omega$ satisfies intcone, and let $b\in L^\infty(Q_\tau)$. Then there exists a unique weak energy solution $\rho_\varepsilon\in W$ to fp, where $W'_\Omega$ being the dual space of $W^{1,2}(\Omega)$, endowed with the standard dual norm. In addition, if $\rho_\tau\geq0$, then $\rho_\varepsilon(t)\geq0$ and we have for $t\in(\tau,T]$.

Theorems & Definitions (14)

  • Theorem 2.1
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.4
  • Remark 3.5
  • proof
  • Remark 3.6
  • Remark 3.7: Towards the $\mathcal{O}(\varepsilon|\log\varepsilon|)$ rate of convergence
  • ...and 4 more