Averaging of Hamilton-Jacobi equations along divergence-free vector fields

Hitoshi Ishii; Taiga Kumagai

doi:10.3934/dcds.2020329

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2021, Volume 41, Issue 4: 1519-1542. Doi: 10.3934/dcds.2020329

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Averaging of Hamilton-Jacobi equations along divergence-free vector fields

Hitoshi Ishii^1, , and
Taiga Kumagai^2,

1.
Institute for Mathematics and Computer Science, Tsuda University, 2-1-1 Tsuda, Kodaira, Tokyo 187-8577 Japan
2.
National Institute of Technology, Maizuru College, 234 Shiroya, Maizuru-shi, Kyoto 625-8511 Japan

^* Corresponding author: Hitoshi Ishii
^* Corresponding author: Hitoshi Ishii

Received: December 31, 2019

Revised: May 31, 2020

Early access: September 2020

Published: April 2021

The first author is partially supported by the JSPS grants: KAKENHI #16H03948, #18H00833, #20K03688, #20H01817. The second author is partially supported by the JSPS grants: KAKENHI #20K03688

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Abstract

We study the asymptotic behavior of solutions to the Dirichlet problem for Hamilton-Jacobi equations with large drift terms, where the drift terms are given by divergence-free vector fields. This is an attempt to understand the averaging effect for fully nonlinear degenerate elliptic equations. In this work, we restrict ourselves to the case of Hamilton-Jacobi equations. The second author has already established averaging results for Hamilton-Jacobi equations with convex Hamiltonians ($ G $ below) under the classical formulation of the Dirichlet condition. Here we treat the Dirichlet condition in the viscosity sense and establish an averaging result for Hamilton-Jacobi equations with relatively general Hamiltonian $ G $.

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Mathematics Subject Classification: Primary: 35B25, 35F21; Secondary: 35F30, 35D40, 49L25.

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Figure 1. $ N = 6 $

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