Periodic homogenization of G-equations and viscosity effects

Yu Yu Liu, Jack Xin, Yifeng Yu

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)

Abstract

G-equations are well-known front propagation models in combustion and are Hamilton-Jacobi type equations with convex but non-coercive Hamiltonians. Viscous G-equations arise from numerical discretization or modeling dissipative mechanisms. Although viscosity helps to overcome non-coercivity, we prove homogenization of an inviscid G-equation based on approximate correctors and attainability of controlled flow trajectories. We verify the attainability for two-dimensional mean zero incompressible flows, and demonstrate asymptotically and numerically that viscosity reduces the homogenized Hamiltonian in cellular flows. In the case of onedimensional compressible flows, we found an explicit formula of homogenized Hamiltonians, as well as necessary and sufficient conditions for wave trapping (effective Hamiltonian vanishes identically). Viscosity restores coercivity and wave propagation.

Original languageEnglish
Pages (from-to)2351-2367
Number of pages17
JournalNonlinearity
Volume23
Issue number10
DOIs
Publication statusPublished - 2010 Oct

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

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