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• W2896526532 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." @default.
• W2896526532 created "2018-10-26" @default.
• W2896526532 creator A5027937292 @default.
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• W2896526532 date "2019-04-04" @default.
• W2896526532 modified "2023-09-27" @default.
• W2896526532 title "Parabolic type equations associated with the Dirichlet form on the Sierpinski gasket" @default.
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• W2896526532 doi "https://doi.org/10.1007/s00440-019-00910-8" @default.
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