Parabolic type equations associated with the Dirichlet form on the Sierpinski gasket
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...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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Springer
2019
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| _version_ | 1797080357860802560 |
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| author | Xuan, L Qian, Z |
| author_facet | Xuan, L Qian, Z |
| author_sort | Xuan, L |
| collection | OXFORD |
| description | 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 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. |
| first_indexed | 2024-03-07T00:58:51Z |
| format | Journal article |
| id | oxford-uuid:89076fb6-e81c-48b7-bcf4-61aa1b50b311 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T00:58:51Z |
| publishDate | 2019 |
| publisher | Springer |
| record_format | dspace |
| spelling | oxford-uuid:89076fb6-e81c-48b7-bcf4-61aa1b50b3112022-03-26T22:21:42ZParabolic type equations associated with the Dirichlet form on the Sierpinski gasketJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:89076fb6-e81c-48b7-bcf4-61aa1b50b311EnglishSymplectic Elements at OxfordSpringer2019Xuan, LQian, ZBy 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 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. |
| spellingShingle | Xuan, L Qian, Z Parabolic type equations associated with the Dirichlet form on the Sierpinski gasket |
| title | Parabolic type equations associated with the Dirichlet form on the Sierpinski gasket |
| title_full | Parabolic type equations associated with the Dirichlet form on the Sierpinski gasket |
| title_fullStr | Parabolic type equations associated with the Dirichlet form on the Sierpinski gasket |
| title_full_unstemmed | Parabolic type equations associated with the Dirichlet form on the Sierpinski gasket |
| title_short | Parabolic type equations associated with the Dirichlet form on the Sierpinski gasket |
| title_sort | parabolic type equations associated with the dirichlet form on the sierpinski gasket |
| work_keys_str_mv | AT xuanl parabolictypeequationsassociatedwiththedirichletformonthesierpinskigasket AT qianz parabolictypeequationsassociatedwiththedirichletformonthesierpinskigasket |