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.

The present note contains a review of p-energies and Sobolev spaces on metric measure spaces that carry a strongly local regular Dirichlet form. These Sobolev spaces are then used to generalize some basic results from the calculus of variations, such as the existence of minimizers for convex functionals and certain constrained mimimization problems. This applies to a number of non-classical situations such as degenerate diffusions, superpositions of diffusions and diffusions on fractals or on products of fractals. Contents 1. Introduction 1 2. Dirichlet forms, p-energies and Sobolev spaces H 1,p 0 (X, m) 3 3. L p -vector fields and reflexivity of Sobolev spaces 5 4. Some examples 8 5. Existence of minimizers for convex functionals 9 6. Some examples 12 7. Constrained minimization problems 12 Appendix A. Uniform convexity of L p 14 Appendix B. Closability of p-energies and distributional gradients 16 Appendix C. Sobolev spaces W 1,p (X, m) 18 References 18

No 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 non-linear partial differential equations on fractal spaces. Contents 1 , 2 Research supported in part by the DFG IRTG 2235: 'Searching for the regular in the irregular: Analysis of singular and random systems'. 1 Research supported in part by the 'Fractal Geometry and Dynamics' program, Institut Mittag-Leffler, Stockholm, 2017, and by the DFG CRC 1283: 'Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications'.

We consider linear partial differential equations on resistance spaces that are uniformly elliptic and parabolic in the sense of quadratic forms and involve abstract gradient and divergence terms. Our main interest is to provide graph and metric graph approximations for their unique solutions. For families of equations with different coefficients on a single compact resistance space we prove that solutions have accumulation points with respect to the uniform convergence in space, provided that the coefficients remain bounded. If in a sequence of equations the coefficients converge suitably, the solutions converge uniformly along a subsequence. For the special case of local resistance forms on finitely ramified sets we also consider sequences of resistance spaces approximating the finitely ramified set from within. Under suitable assumptions on the coefficients (extensions of) linearizations of the solutions of equations on the approximating spaces accumulate or even converge uniformly along a subsequence to the solution of the target equation on the finitely ramified set. The results cover discrete and metric graph approximations, and both are discussed.