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Homogenization of Weakly Coupled Systems of Hamilton–Jacobi Equations with Fast Switching Rates

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Abstract

We consider homogenization for weakly coupled systems of Hamilton–Jacobi equations with fast switching rates. The fast switching rate terms force the solutions to converge to the same limit, which is a solution of the effective equation. We discover the appearance of the initial layers, which appear naturally when we consider the systems with different initial data and analyze them rigorously. In particular, we obtain matched asymptotic solutions of the systems and the rate of convergence. We also investigate properties of the effective Hamiltonian of weakly coupled systems and show some examples which do not appear in the context of single equations.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Science, Fukuoka University, Fukuoka, 814-0180, Japan

    Hiroyoshi Mitake

  2. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    Hung V. Tran

Authors
  1. Hiroyoshi Mitake
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  2. Hung V. Tran
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Correspondence to Hung V. Tran.

Additional information

Communicated by F. Lin

Dedicated to Professor H. Ishii on the occasion of his 65th birthday

This work was partially done while the first author visited the Mathematics Department at the University of California, Berkeley.

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Mitake, H., Tran, H.V. Homogenization of Weakly Coupled Systems of Hamilton–Jacobi Equations with Fast Switching Rates. Arch Rational Mech Anal 211, 733–769 (2014). https://doi.org/10.1007/s00205-013-0685-x

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  • DOI: https://doi.org/10.1007/s00205-013-0685-x

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