Analysis of a dilute polymer model with a time-fractional derivative
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| Format: | Preprint |
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2023
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| _version_ | 1866914458674135040 |
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| author | Fritz, Marvin Süli, Endre Wohlmuth, Barbara |
| author_facet | Fritz, Marvin Süli, Endre Wohlmuth, Barbara |
| contents | We investigate the well-posedness of a coupled Navier-Stokes-Fokker-Planck system with a time-fractional derivative. Such systems arise in the kinetic theory of dilute solutions of polymeric liquids, where the motion of noninteracting polymer chains in a Newtonian solvent is modelled by a stochastic process exhibiting power-law waiting time, in order to capture subdiffusive processes associated with non-Fickian diffusion. We outline the derivation of the model from a subordinated Langevin equation. The elastic properties of the polymer molecules immersed in the solvent are modelled by a finitely extensible nonlinear elastic (FENE) dumbbell model, and the drag term in the Fokker--Planck equation is assumed to be corotational. We prove the global-in-time existence of large-data weak solutions to this time-fractional model of order $α\in (\tfrac12,1)$, and derive an energy inequality satisfied by weak solutions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_16606 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Analysis of a dilute polymer model with a time-fractional derivative Fritz, Marvin Süli, Endre Wohlmuth, Barbara Analysis of PDEs 35Q30, 35Q84, 35R11, 60G22, 82C31, 82D60 We investigate the well-posedness of a coupled Navier-Stokes-Fokker-Planck system with a time-fractional derivative. Such systems arise in the kinetic theory of dilute solutions of polymeric liquids, where the motion of noninteracting polymer chains in a Newtonian solvent is modelled by a stochastic process exhibiting power-law waiting time, in order to capture subdiffusive processes associated with non-Fickian diffusion. We outline the derivation of the model from a subordinated Langevin equation. The elastic properties of the polymer molecules immersed in the solvent are modelled by a finitely extensible nonlinear elastic (FENE) dumbbell model, and the drag term in the Fokker--Planck equation is assumed to be corotational. We prove the global-in-time existence of large-data weak solutions to this time-fractional model of order $α\in (\tfrac12,1)$, and derive an energy inequality satisfied by weak solutions. |
| title | Analysis of a dilute polymer model with a time-fractional derivative |
| topic | Analysis of PDEs 35Q30, 35Q84, 35R11, 60G22, 82C31, 82D60 |
| url | https://arxiv.org/abs/2307.16606 |