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 $$L^{p}$$Lp 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.

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与谢尔宾斯基垫片上的狄利克雷形式相关的抛物线型方程

通过使用来自随机分析的分析工具，我们开始研究谢尔宾斯基垫片上的一些非线性抛物线方程，其动机是流体沿分形流动的建模（可被视为简化的粗糙多孔介质模型）。与常规空间情况不同，这种涉及非线性对流项的抛物线型方程必须采用不同的形式，因为对流项对于定义热半群的“线性部分”必须是奇异的。为了研究这些抛物型方程，将建立一种新型的垫圈上 Dirichlet 形式的 Sobolev 不等式。这些 Sobolev 不等式本身就很有趣，并且与欧几里得空间的情况相反，涉及两个关于两个相互奇异的测度的 $$L^{p}$$Lp 范数。通过检查相关热半群奇异卷积的性质，我们推导出了这些抛物线方程在一些技术条件下解的时空规律。还研究了Sierpinski垫片上的Burgers方程，利用向后随机微分方程的技术推导出了解的极大值原理，得到了其解的存在性、唯一性和正则性。