To blow-up or not to blow-up for a granular kinetic equation
A simplified kinetic description of rapid granular media leads to a nonlocal Vlasov-type equation with a convolution integral operator that is of the same form as the continuity equations for aggregation-diffusion macroscopic dynamics. While the singular behavior of these nonlinear continuity equati...
| Main Authors: | , , , |
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| Format: | Journal article |
| Language: | English |
| Published: |
Elsevier
2024
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| _version_ | 1826315147058085888 |
|---|---|
| author | Carrillo, JA Shu, R Wang, L Xu, W |
| author_facet | Carrillo, JA Shu, R Wang, L Xu, W |
| author_sort | Carrillo, JA |
| collection | OXFORD |
| description | A simplified kinetic description of rapid granular media leads to a nonlocal
Vlasov-type equation with a convolution integral operator that is of the same
form as the continuity equations for aggregation-diffusion macroscopic
dynamics. While the singular behavior of these nonlinear continuity equations
is well studied in the literature, the extension to the corresponding granular
kinetic equation is highly nontrivial. The main question is whether the
singularity formed in velocity direction will be enhanced or mitigated by the
shear in phase space due to free transport. We present a preliminary study
through a meticulous numerical investigation and heuristic arguments. We have
numerically developed a structure-preserving method with adaptive mesh
refinement that can effectively capture potential blow-up behavior in the
solution for granular kinetic equations. We have analytically constructed a
finite-time blow-up infinite mass solution and discussed how this can provide
insights into the finite mass scenario. |
| first_indexed | 2024-12-09T03:21:17Z |
| format | Journal article |
| id | oxford-uuid:97f4de83-ac62-44f6-8522-aab7d4b6205c |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-12-09T03:21:17Z |
| publishDate | 2024 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | oxford-uuid:97f4de83-ac62-44f6-8522-aab7d4b6205c2024-11-12T08:54:40ZTo blow-up or not to blow-up for a granular kinetic equationJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:97f4de83-ac62-44f6-8522-aab7d4b6205cEnglishSymplectic ElementsElsevier2024Carrillo, JAShu, RWang, LXu, WA simplified kinetic description of rapid granular media leads to a nonlocal Vlasov-type equation with a convolution integral operator that is of the same form as the continuity equations for aggregation-diffusion macroscopic dynamics. While the singular behavior of these nonlinear continuity equations is well studied in the literature, the extension to the corresponding granular kinetic equation is highly nontrivial. The main question is whether the singularity formed in velocity direction will be enhanced or mitigated by the shear in phase space due to free transport. We present a preliminary study through a meticulous numerical investigation and heuristic arguments. We have numerically developed a structure-preserving method with adaptive mesh refinement that can effectively capture potential blow-up behavior in the solution for granular kinetic equations. We have analytically constructed a finite-time blow-up infinite mass solution and discussed how this can provide insights into the finite mass scenario. |
| spellingShingle | Carrillo, JA Shu, R Wang, L Xu, W To blow-up or not to blow-up for a granular kinetic equation |
| title | To blow-up or not to blow-up for a granular kinetic equation |
| title_full | To blow-up or not to blow-up for a granular kinetic equation |
| title_fullStr | To blow-up or not to blow-up for a granular kinetic equation |
| title_full_unstemmed | To blow-up or not to blow-up for a granular kinetic equation |
| title_short | To blow-up or not to blow-up for a granular kinetic equation |
| title_sort | to blow up or not to blow up for a granular kinetic equation |
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