Measure solutions to a system of continuity equations driven by Newtonian nonlocal interactions
We prove global-in-time existence and uniqueness of measure solutions of a nonlocal interaction system of two species in one spatial dimension. For initial data including atomic parts we provide a notion of gradient-flow solutions in terms of the pseudo-inverses of the corresponding cumulative distr...
| Main Authors: | , , , , |
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| Format: | Journal article |
| Language: | English |
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American Institute of Mathematical Sciences
2019
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| _version_ | 1826300367784116224 |
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| author | Carrillo, JA Di Francesco, M Esposito, A Fagioli, S Schmidtchen, M |
| author_facet | Carrillo, JA Di Francesco, M Esposito, A Fagioli, S Schmidtchen, M |
| author_sort | Carrillo, JA |
| collection | OXFORD |
| description | We prove global-in-time existence and uniqueness of measure solutions of a nonlocal interaction system of two species in one spatial dimension. For initial data including atomic parts we provide a notion of gradient-flow solutions in terms of the pseudo-inverses of the corresponding cumulative distribution functions, for which the system can be stated as a gradient flow on the Hilbert space L2(0, 1)2 according to the classical theory by Brézis. For absolutely continuous initial data we construct solutions using a minimising movement scheme in the set of probability measures. In addition we show that the scheme preserves finiteness of the Lm-norms for all m ∈ [1, +∞] and of the second moments. We then provide a characterisation of equilibria and prove that they are achieved (up to time subsequences) in the large time asymptotics. We conclude the paper constructing two examples of non-uniqueness of measure solutions emanating from the same (atomic) initial datum, showing that the notion of gradient flow solution is necessary to single out a unique measure solution. |
| first_indexed | 2024-03-07T05:16:05Z |
| format | Journal article |
| id | oxford-uuid:dd3cb088-d901-4a48-b016-0d2a91d14aa3 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T05:16:05Z |
| publishDate | 2019 |
| publisher | American Institute of Mathematical Sciences |
| record_format | dspace |
| spelling | oxford-uuid:dd3cb088-d901-4a48-b016-0d2a91d14aa32022-03-27T09:23:42ZMeasure solutions to a system of continuity equations driven by Newtonian nonlocal interactionsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:dd3cb088-d901-4a48-b016-0d2a91d14aa3EnglishSymplectic ElementsAmerican Institute of Mathematical Sciences2019Carrillo, JADi Francesco, MEsposito, AFagioli, SSchmidtchen, MWe prove global-in-time existence and uniqueness of measure solutions of a nonlocal interaction system of two species in one spatial dimension. For initial data including atomic parts we provide a notion of gradient-flow solutions in terms of the pseudo-inverses of the corresponding cumulative distribution functions, for which the system can be stated as a gradient flow on the Hilbert space L2(0, 1)2 according to the classical theory by Brézis. For absolutely continuous initial data we construct solutions using a minimising movement scheme in the set of probability measures. In addition we show that the scheme preserves finiteness of the Lm-norms for all m ∈ [1, +∞] and of the second moments. We then provide a characterisation of equilibria and prove that they are achieved (up to time subsequences) in the large time asymptotics. We conclude the paper constructing two examples of non-uniqueness of measure solutions emanating from the same (atomic) initial datum, showing that the notion of gradient flow solution is necessary to single out a unique measure solution. |
| spellingShingle | Carrillo, JA Di Francesco, M Esposito, A Fagioli, S Schmidtchen, M Measure solutions to a system of continuity equations driven by Newtonian nonlocal interactions |
| title | Measure solutions to a system of continuity equations driven by Newtonian nonlocal interactions |
| title_full | Measure solutions to a system of continuity equations driven by Newtonian nonlocal interactions |
| title_fullStr | Measure solutions to a system of continuity equations driven by Newtonian nonlocal interactions |
| title_full_unstemmed | Measure solutions to a system of continuity equations driven by Newtonian nonlocal interactions |
| title_short | Measure solutions to a system of continuity equations driven by Newtonian nonlocal interactions |
| title_sort | measure solutions to a system of continuity equations driven by newtonian nonlocal interactions |
| work_keys_str_mv | AT carrilloja measuresolutionstoasystemofcontinuityequationsdrivenbynewtoniannonlocalinteractions AT difrancescom measuresolutionstoasystemofcontinuityequationsdrivenbynewtoniannonlocalinteractions AT espositoa measuresolutionstoasystemofcontinuityequationsdrivenbynewtoniannonlocalinteractions AT fagiolis measuresolutionstoasystemofcontinuityequationsdrivenbynewtoniannonlocalinteractions AT schmidtchenm measuresolutionstoasystemofcontinuityequationsdrivenbynewtoniannonlocalinteractions |