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Convergence & rates for Hamilton–Jacobi equations with Kirchoff junction conditions

  • Peter S. MorfeEmail author
Article
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Abstract

We investigate rates of convergence for two approximation schemes of time-independent and time-dependent Hamilton–Jacobi equations with Kirchoff junction conditions. We analyze the vanishing viscosity limit and monotone finite-difference schemes. Following recent work of Lions and Souganidis, we impose no convexity assumptions on the Hamiltonians. For stationary Hamilton–Jacobi equations, we obtain the classical \(\epsilon ^{\frac{1}{2}}\) rate, while we obtain an \(\epsilon ^{\frac{1}{7}}\) rate for approximations of the Cauchy problem. In addition, we present a number of new techniques of independent interest, including a quantified comparison proof for the Cauchy problem and an equivalent definition of the Kirchoff junction condition.

Keywords

Hamilton–Jacobi equations Junction problems Stratification problems Vanishing viscosity limit Monotone finite difference schemes 

Mathematics Subject Classification

35F20 65N12 65M12 

Notes

Acknowledgements

It is a pleasure to acknowledge P.E. Souganidis for suggesting this problem and for enlightening discussions. Credit is due as well to the anonymous reviewers for their sage advice and for pointing out a number of typos, and to M. Sardarli for helpful comments. The author was partially supported by the National Science Foundation Research Training Group Grant DMS-1246999.

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA

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