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

Xuan Liu, Zhongmin Qian

Abstract

By using the analytic tools of Dirichlet forms, we initiate a study of some non-linear parabolic equations on Sierpinski gasket, motivated by modellings of fluid flows along a fractal (which can be considered as a simplified rough porous medium). Unlike the regular spaces case, such equation involving a convection term must take a quite different form and the convection term must be singular to the "linear part" which is determined 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 as an effective tool for our study. These Sobolev inequalities, which are interesting by their own and in contrast to the case of Euclidean spaces, involve two $L^p$ norms with respect two mutually singular measures. By examining properties of a singular convolution of the associated heat semigroup, we derive the space-time regularity of solutions to the 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 and uniqueness as well as the regularity of solutions are obtained.

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

Abstract

By using the analytic tools of Dirichlet forms, we initiate a study of some non-linear parabolic equations on Sierpinski gasket, motivated by modellings of fluid flows along a fractal (which can be considered as a simplified rough porous medium). Unlike the regular spaces case, such equation involving a convection term must take a quite different form and the convection term must be singular to the "linear part" which is determined 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 as an effective tool for our study. These Sobolev inequalities, which are interesting by their own and in contrast to the case of Euclidean spaces, involve two norms with respect two mutually singular measures. By examining properties of a singular convolution of the associated heat semigroup, we derive the space-time regularity of solutions to the 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 and uniqueness as well as the regularity of solutions are obtained.

Paper Structure

This paper contains 5 sections, 18 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sobolev inequalities
  4. Semi-linear parabolic PDEs
  5. The Burgers equations

Key Result

Theorem 3.6

Let $1\le p\le q\le\infty,\;q\ge2$. Suppose $\hat{\sigma}$ is a Borel measure on $\hat{\mathbb{S}}$ satisfying the condition eq:-47. Then where and $C>0$ is a constant depending only on the constant $C_{\hat{\sigma}}$ in eq:-47. Moreover, if there exists a sequence $\{S_{m}\}_{m\in\mathbb{Z}}$ of dyadic triangles such that and then the pair of exponents given by eq:-6 is optimal in the followi

Theorems & Definitions (48)

  • Remark 2.1
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Remark 3.4
  • Example 3.5
  • Theorem 3.6
  • proof
  • Remark 3.7
  • Theorem 3.8
  • ...and 38 more