Parabolic type equations associated with the Dirichlet form on the Sierpinski gasket

Xuan Liu,Zhongmin Qian

doi:10.1007/s00440-019-00910-8

Xuan Liu, Zhongmin Qian

Open Access

https://doi.org/10.1007/s00440-019-00910-8

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Abstract

By using analytic tools from stochastic analysis, we initiate a study of some non-linear parabolic equations on Sierpinski gasket, motivated by modellings of fluid flows along fractals (which can be considered as models of simplified rough porous media). Unlike the regular space case, such parabolic type equations involving non-linear convection terms must take a different form, due to the fact that convection terms must be singular to the “linear part” which defines the heat semigroup. In order to study these parabolic type equations, a new kind of Sobolev inequalities for the Dirichlet form on the gasket will be established. These Sobolev inequalities, which are interesting on their own and in contrast to the case of Euclidean spaces, involve two L^{p} norms with respect to two mutually singular measures. By examining properties of singular convolutions of the associated heat semigroup, we derive the space-time regularity of solutions to these parabolic equations under a few technical conditions. The Burgers equations on the Sierpinski gasket are also studied, for which a maximum principle for solutions is derived using techniques from backward stochastic differential equations, and the existence, uniqueness, and regularity of its solutions are obtained.

Highlights

The analysis on fractals has attracted attentions of researchers in the last decades, for the reason that fractals are archetypal examples of spaces without suitable smooth structure, and because fractals are examples of interesting models in statistical mechanics
Since a calculus on fractals is not available, the theory of Dirichlet forms on measure-metric spaces and stochastic calculus are the analytic tools employed for the study of analysis problems on fractals, and many interesting results have been established in the past decades
Brownian motion on the Sierpinski gasket was first constructed by Goldstein and Kusuoka as the limit of a sequence of random walks on lattices

Summary

Introduction

The analysis on fractals has attracted attentions of researchers in the last decades, for the reason that fractals are archetypal examples of spaces without suitable smooth structure, and because fractals are examples of interesting models in statistical mechanics. The construction of gradients of functions with finite energy has been given in Kusuoka in [25], where a significant difference between Euclidean spaces and fractals has been revealed (see [25, Section 6]). By virtue of the results obtained in [25], gradients of functions on the Sierpinski gasket may be defined as square integrable functions with respect to Kusuoka’s measure A new class of semi-linear parabolic equations involving singular measures on the Sierpinski gasket was proposed and studied in [27], where, among other things, a Feynman–Kac representation was obtained assuming the existence of weak solutions. A crucial ingredient in our argument is a new type of Sobolev inequalities on the Sierpinski gasket (and the infinite gasket) involving different measures (which can be mutually singular). The main results and the arguments given in this paper can be adapted without difficulties

Preliminaries

Sobolev inequalities

Semi-linear parabolic PDEs

The Burgers equations