The Filippov characteristic flow for the aggregation equation with mildly singular potentials
Existence and uniqueness of global in time measure solution for the multidimensional aggregation equation is analyzed. Such a system can be written as a continuity equation with a velocity field computed through a self-consistent interaction potential. In Carrillo et al. (2011) [17], a well-posednes...
| Main Authors: | , , , |
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| Format: | Journal article |
| Language: | English |
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Elsevier
2015
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| _version_ | 1826302837757313024 |
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| author | Carrillo de la Plata, JA James, F Lagoutiere, F Vauchelet, N |
| author_facet | Carrillo de la Plata, JA James, F Lagoutiere, F Vauchelet, N |
| author_sort | Carrillo de la Plata, JA |
| collection | OXFORD |
| description | Existence and uniqueness of global in time measure solution for the multidimensional aggregation equation is analyzed. Such a system can be written as a continuity equation with a velocity field computed through a self-consistent interaction potential. In Carrillo et al. (2011) [17], a well-posedness theory based on the geometric approach of gradient flows in measure metric spaces has been developed for mildly singular potentials at the origin under the basic assumption of being λ-convex. We propose here an alternative method using classical tools from PDEs. We show the existence of a characteristic flow based on Filippov's theory of discontinuous dynamical systems such that the weak measure solution is the pushforward measure with this flow. Uniqueness is obtained thanks to a contraction argument in transport distances using the λ-convexity of the potential. Moreover, we show the equivalence of this solution with the gradient flow solution. Finally, we show the convergence of a numerical scheme for general measure solutions in this framework allowing for the simulation of solutions for initial smooth densities after their first blow-up time in Lp-norms. |
| first_indexed | 2024-03-07T05:53:29Z |
| format | Journal article |
| id | oxford-uuid:e9b61117-5fed-4496-acf7-b0f6dffbaac6 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T05:53:29Z |
| publishDate | 2015 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | oxford-uuid:e9b61117-5fed-4496-acf7-b0f6dffbaac62022-03-27T10:56:13ZThe Filippov characteristic flow for the aggregation equation with mildly singular potentialsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:e9b61117-5fed-4496-acf7-b0f6dffbaac6EnglishSymplectic ElementsElsevier 2015Carrillo de la Plata, JAJames, FLagoutiere, FVauchelet, NExistence and uniqueness of global in time measure solution for the multidimensional aggregation equation is analyzed. Such a system can be written as a continuity equation with a velocity field computed through a self-consistent interaction potential. In Carrillo et al. (2011) [17], a well-posedness theory based on the geometric approach of gradient flows in measure metric spaces has been developed for mildly singular potentials at the origin under the basic assumption of being λ-convex. We propose here an alternative method using classical tools from PDEs. We show the existence of a characteristic flow based on Filippov's theory of discontinuous dynamical systems such that the weak measure solution is the pushforward measure with this flow. Uniqueness is obtained thanks to a contraction argument in transport distances using the λ-convexity of the potential. Moreover, we show the equivalence of this solution with the gradient flow solution. Finally, we show the convergence of a numerical scheme for general measure solutions in this framework allowing for the simulation of solutions for initial smooth densities after their first blow-up time in Lp-norms. |
| spellingShingle | Carrillo de la Plata, JA James, F Lagoutiere, F Vauchelet, N The Filippov characteristic flow for the aggregation equation with mildly singular potentials |
| title | The Filippov characteristic flow for the aggregation equation with mildly singular potentials |
| title_full | The Filippov characteristic flow for the aggregation equation with mildly singular potentials |
| title_fullStr | The Filippov characteristic flow for the aggregation equation with mildly singular potentials |
| title_full_unstemmed | The Filippov characteristic flow for the aggregation equation with mildly singular potentials |
| title_short | The Filippov characteristic flow for the aggregation equation with mildly singular potentials |
| title_sort | filippov characteristic flow for the aggregation equation with mildly singular potentials |
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