On the viscous Burgers equation on metric graphs and fractals

Abstract

We study a formulation of Burgers equation on spaces carrying a local regular resistance form in the sense of Kigami. 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. On metric graphs the difference between these two equations can be understood in terms of different vertex conditions. For the second formulation we show existence and uniqueness of solutions and verify the continuous dependence on the initial condition. For resistance forms associated with regular harmonic structures on p.c.f. self-similar sets we also prove that solutions 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 non-linear partial differential equations on fractal spaces.