On the viscous Burgers equation on metric graphs and fractals

We study a formulation of Burgers equation on the Sierpinski gasket, which is the prototype of a p.c.f. self-similar fractal. One possibility is to implement Burgers equation as a semilinear heat equation associated with the Laplacian for scalar functions, just as on the unit interval. Here we propo...

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Bibliographic Details
Published inarXiv.org
Main Authors Hinz, Michael, Meinert, Melissa
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 14.12.2017
Subjects
Burgers equation
Dependence
Fields (mathematics)
Fractals
Graphs
Linear equations
Mathematical analysis
Mathematics - Analysis of PDEs
Mathematics - Functional Analysis
Mathematics - Mathematical Physics
Nonlinear differential equations
Nonlinear equations
Partial differential equations
Physics - Mathematical Physics
Self-similarity
Thermodynamics
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Summary:We study a formulation of Burgers equation on the Sierpinski gasket, which is the prototype of a p.c.f. self-similar fractal. One possibility is to implement Burgers equation as a semilinear heat equation associated with the Laplacian for scalar functions, just as on the unit interval. Here we propose a second, different formulation which follows from the Cole-Hopf transform and is associated with the Laplacian for vector fields. The difference between these two equations can be understood in terms of different vertex conditions for Laplacians on metric graphs. For the second formulation we show existence and uniqueness of solutions and verify the continuous dependence on the initial condition. We also prove that solutions on the Sierpinski gasket can be approximated in a weak sense by solutions to corresponding equations on approximating metric graphs. These results are part of a larger program discussing nonlinear partial differential equations on fractal spaces.
ISSN:2331-8422
DOI:10.48550/arxiv.1712.05472