On the viscous Burgers equation on metric graphs and fractals

TL;DR

This paper explores two formulations of the viscous Burgers equation on the Sierpinski gasket, establishing existence, uniqueness, and approximation of solutions, thereby advancing the understanding of nonlinear PDEs on fractal spaces.

Contribution

It introduces a novel formulation of Burgers equation on fractals based on vector Laplacians and vertex conditions, with proofs of solution properties and approximation methods.

Findings

Existence and uniqueness of solutions for the vector field formulation.

Solutions can be approximated by equations on metric graph sequences.

Different vertex conditions lead to distinct formulations of the equation.

Abstract

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.