# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2020329

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

 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

Received  January 2020 Revised  June 2020 Published  September 2020

Fund Project: 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

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 $G$.

Citation: Hitoshi Ishii, Taiga Kumagai. Averaging of Hamilton-Jacobi equations along divergence-free vector fields. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020329
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$N = 6$
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