Vanishing viscosity approximations of possibly degenerate viscous Hamilton–Jacobi equations on the torus converge pointwise at rate O(ε|log ε|) and in averaged form at rate O(ε).
A new selection problem for degenerate viscous Hamilton-Jacobi equations
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abstract
We study a selection problem for degenerate viscous Hamilton--Jacobi equations with convex Hamiltonians, in which the approximation procedure combines a nonlinear discounted approximation with a small potential perturbation. A key question is how their simultaneous effects influence the asymptotic selection of viscosity solutions of the associated ergodic problem. Based on the nonlinear adjoint method, we establish the uniform convergence of the approximating solutions to a distinguished solution of the ergodic problem and derive a formula for the selected limit in terms of generalized Mather measures and the potential. As an application, we show that this selection principle is sufficiently flexible to realize any prescribed solution of the ergodic problem, with an explicit convergence rate.
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Convergence Rates for Vanishing Viscosity Approximations of Possibly Degenerate Viscous Hamilton--Jacobi Equations
Vanishing viscosity approximations of possibly degenerate viscous Hamilton–Jacobi equations on the torus converge pointwise at rate O(ε|log ε|) and in averaged form at rate O(ε).