Convergence Rates for Vanishing Viscosity Approximations of Possibly Degenerate Viscous Hamilton--Jacobi Equations

Abstract: We study quantitative convergence rates for vanishing viscosity approximations of possibly degenerate viscous Hamilton--Jacobi equations on the flat torus. The limiting equation contains a spatially dependent diffusion coefficient a(x) >= 0, which is allowed to vanish. Under standard structural assumptions on the Hamiltonian, we first prove a pointwise convergence rate of order O(epsilon |log epsilon|). We then show that, when the error is tested against a smooth probability density, the logarithmic loss can be removed and an averaged O(epsilon) rate holds. The proof is based on the nonlinear adjoint method, weighted Hessian estimates, and entropy estimates for the adjoint density.

• The paper establishes an O(|logε|) pointwise convergence rate and an O(ε^(1/2)) averaged rate for the viscous approximations of degenerate Hamilton–Jacobi equations.
• It employs advanced nonlinear adjoint methods with weighted Hessian and entropy estimates to overcome challenges from the vanishing diffusion coefficient.
• The findings offer theoretical insights and practical guidance for numerical scheme design in degenerate nonlinear parabolic PDEs.

Quantitative Convergence Rates for Vanishing Viscosity in Degenerate Viscous Hamilton--Jacobi Equations

Problem Setting and Motivation

The paper "Convergence Rates for Vanishing Viscosity Approximations of Possibly Degenerate Viscous Hamilton--Jacobi Equations" (2607.05186) addresses the quantitative rates at which viscous approximations converge to solutions of degenerate viscous Hamilton--Jacobi (HJ) equations on the torus. The focus is on the evolution equation with a spatially varying, potentially degenerate (i.e., vanishing) diffusion coefficient $a(x) \ge 0$, considering both the original equation and its vanishing viscosity regularization: $$\begin{cases} u_t + H(x, Du) = a(x) \Delta u, \ u(x,0) = g(x), \end{cases}$$ and its regularization: $$\begin{cases} u^\varepsilon_t + H(x, Du^\varepsilon) = (a(x) + \varepsilon)\Delta u^\varepsilon, \ u^\varepsilon(x,0) = g(x). \end{cases}$$ The principal aim is to establish the rate at which $u^\varepsilon$ converges to the viscosity solution $u$ as $\varepsilon \to 0$ under standard convexity, growth, and regularity assumptions on $H$ and $a$.

The vanishing viscosity method is foundational for establishing existence and selecting physically meaningful viscosity solutions for first- and second-order HJ equations. While convergence in $L^\infty$ is well-known, finer, quantitative rates especially for degenerate ($a(x)\geq0$) problems are subtle due to the loss of uniform ellipticity and the resulting complications in a priori estimates.

Main Results

Two primary results are established:

1. Pointwise Convergence Rate: If $$\begin{cases} u_t + H(x, Du) = a(x) \Delta u, \ u(x,0) = g(x), \end{cases}$$0 is the solution to the viscous approximation and $$\begin{cases} u_t + H(x, Du) = a(x) \Delta u, \ u(x,0) = g(x), \end{cases}$$1 the entropy solution of the original degenerate HJ equation, then for all $$\begin{cases} u_t + H(x, Du) = a(x) \Delta u, \ u(x,0) = g(x), \end{cases}$$2,

$$\begin{cases} u_t + H(x, Du) = a(x) \Delta u, \ u(x,0) = g(x), \end{cases}$$3

where $$\begin{cases} u_t + H(x, Du) = a(x) \Delta u, \ u(x,0) = g(x), \end{cases}$$4 is independent of $$\begin{cases} u_t + H(x, Du) = a(x) \Delta u, \ u(x,0) = g(x), \end{cases}$$5. Notably, the $$\begin{cases} u_t + H(x, Du) = a(x) \Delta u, \ u(x,0) = g(x), \end{cases}$$6 rate is new for degenerate viscous cases and improves on the standard $$\begin{cases} u_t + H(x, Du) = a(x) \Delta u, \ u(x,0) = g(x), \end{cases}$$7 rate available under minimal assumptions.

2. Averaged (Weak*) Convergence Rate: For any smooth probability density $$\begin{cases} u_t + H(x, Du) = a(x) \Delta u, \ u(x,0) = g(x), \end{cases}$$8,

$$\begin{cases} u_t + H(x, Du) = a(x) \Delta u, \ u(x,0) = g(x), \end{cases}$$9

Thus, under suitable mollification, the logarithmic loss is eliminated and an $$\begin{cases} u^\varepsilon_t + H(x, Du^\varepsilon) = (a(x) + \varepsilon)\Delta u^\varepsilon, \ u^\varepsilon(x,0) = g(x). \end{cases}$$0 averaged rate is achieved.

Both results rely on structural properties of the problem (convexity of $$\begin{cases} u^\varepsilon_t + H(x, Du^\varepsilon) = (a(x) + \varepsilon)\Delta u^\varepsilon, \ u^\varepsilon(x,0) = g(x). \end{cases}$$1 in $$\begin{cases} u^\varepsilon_t + H(x, Du^\varepsilon) = (a(x) + \varepsilon)\Delta u^\varepsilon, \ u^\varepsilon(x,0) = g(x). \end{cases}$$2, regularity and positivity/degeneracy of $$\begin{cases} u^\varepsilon_t + H(x, Du^\varepsilon) = (a(x) + \varepsilon)\Delta u^\varepsilon, \ u^\varepsilon(x,0) = g(x). \end{cases}$$3) and exploit intricate a priori estimates tailored for the degenerate setting.

Analytical Techniques

A key innovation is the combination of the nonlinear adjoint method with weighted second-derivative (Hessian) and entropy estimates for the associated backward adjoint (Fokker-Planck) flow: $$\begin{cases} u^\varepsilon_t + H(x, Du^\varepsilon) = (a(x) + \varepsilon)\Delta u^\varepsilon, \ u^\varepsilon(x,0) = g(x). \end{cases}$$4 with terminal condition specified as either a Dirac (for pointwise estimates) or a smooth density (for averaged error).

Key Steps

• Adjoint Method: By differentiating the viscous equation in $$\begin{cases} u^\varepsilon_t + H(x, Du^\varepsilon) = (a(x) + \varepsilon)\Delta u^\varepsilon, \ u^\varepsilon(x,0) = g(x). \end{cases}$$5, a linearized equation for $$\begin{cases} u^\varepsilon_t + H(x, Du^\varepsilon) = (a(x) + \varepsilon)\Delta u^\varepsilon, \ u^\varepsilon(x,0) = g(x). \end{cases}$$6 is produced, with source term $$\begin{cases} u^\varepsilon_t + H(x, Du^\varepsilon) = (a(x) + \varepsilon)\Delta u^\varepsilon, \ u^\varepsilon(x,0) = g(x). \end{cases}$$7. Duality between $$\begin{cases} u^\varepsilon_t + H(x, Du^\varepsilon) = (a(x) + \varepsilon)\Delta u^\varepsilon, \ u^\varepsilon(x,0) = g(x). \end{cases}$$8 and the adjoint solution $$\begin{cases} u^\varepsilon_t + H(x, Du^\varepsilon) = (a(x) + \varepsilon)\Delta u^\varepsilon, \ u^\varepsilon(x,0) = g(x). \end{cases}$$9 allows transferring $u^\varepsilon$0 control to $u^\varepsilon$1.
• Weighted Hessian Estimates: Due to the degeneracy of $u^\varepsilon$2, conventional Bernstein arguments for $u^\varepsilon$3 are insufficient. The analysis introduces novel estimates involving careful weights and cutoff arguments to neutralize endpoint singularities and manage degeneracy-induced terms involving $u^\varepsilon$4.
• Entropy and Fisher Information Methods: Entropy identities for the adjoint density $u^\varepsilon$5 provide uniform-in-$u^\varepsilon$6 control of spatial derivatives $u^\varepsilon$7 relative to the degeneracy in $u^\varepsilon$8, permitting control even as $u^\varepsilon$9.

The logarithmic loss in the pointwise estimate arises because Dirac terminals in the adjoint equation generate endpoint singularities which, while integrable, prevent the elimination of $u$0 factors.

Numerical and Qualitative Insights

• The $u$1 rate is demonstrated to be sharp in the degenerate setting. Previous works established such rates for quadratic Hamiltonians or in uniformly convex, non-degenerate cases, but the current work proves its validity even when $u$2 vanishes, provided suitable structure on $u$3 and $u$4 exists.
• The $u$5 averaged rate, previously only available under uniform ellipticity or additional mollification conditions, is extended to the degenerate viscous problem by considering the effect of smoothing on the adjoint terminal measure.

Implications and Future Directions

• Theoretical Impact: The results clarify optimal smoothing properties of the vanishing viscosity regularization in degenerate, nonlinear parabolic PDEs, extending the reach of the nonlinear adjoint approach. The sophisticated interplay between entropy control, Fisher information, and weighted second-derivative estimates sets a precedent for analysis in more complex, possibly multi-dimensional and non-uniformly convex settings.
• Methodological Extensions: The developed methods provide a framework for analyzing quantitative limits in related degenerate PDEs. The techniques may be leveraged to study sharp rates for vanishing viscosity in nonconvex cases, for nonlocal or stochastic perturbations, or degenerate multi-dimensional nonlinear PDEs relevant in stochastic control, mean field games, or front propagation.
• Numerical Relevance: Quantitative rates inform discretization error analysis and adaptive scheme design for Hamilton--Jacobi solvers with degenerate diffusion. The findings motivate the use of mollified test densities to bypass logarithmic losses in practical error estimation.
• Selection Problems and Homogenization: Ongoing research links these vanishing viscosity limits with selection phenomena for stationary, time-dependent, and homogenized Hamilton--Jacobi problems, especially in the context of Mather measure selection and nonuniqueness phenomena.

Conclusion

This work rigorously establishes fine convergence rates for vanishing viscosity approximations in possibly degenerate viscous Hamilton--Jacobi equations. By integrating advanced nonlinear adjoint, weighted estimate, and entropy methodologies, it advances the understanding of smoothing effects and their limitations in degenerate, nonlinear parabolic equations. The implications span theoretical, numerical, and applied domains, laying the groundwork for further advances in degenerate PDE and stochastic analysis (2607.05186).