Stability of equilibria of the spatially inhomogeneous Vicsek-BGK equation across a bifurcation
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917850613022720 |
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| author | Merino-Aceituno, Sara Schmeiser, Christian Winter, Raphael |
| author_facet | Merino-Aceituno, Sara Schmeiser, Christian Winter, Raphael |
| contents | The Vicsek-BGK equation is a kinetic model for alignment of particles moving with constant speed between stochastic reorientation events with sampling from a von Mises distribution. The spatially homogeneous model shows a steady state bifurcation with exchange of stability. The main result of this work is an extension of the bifurcation result to the spatially inhomogeneous problem under the additional assumption of a sufficiently large Knudsen number. The mathematical core is the proof of linearized stability, which employs a new hypocoercivity approach based on Laplace-Fourier transformation. The bifurcation result includes global existence of smooth solutions for close-to-equilibrium initial data. For large data smooth solutions might blow up in finite time whereas weak solutions with bounded Boltzmann entropy are shown to exist globally. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_00347 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stability of equilibria of the spatially inhomogeneous Vicsek-BGK equation across a bifurcation Merino-Aceituno, Sara Schmeiser, Christian Winter, Raphael Analysis of PDEs Adaptation and Self-Organizing Systems 35Q92, 35B32 The Vicsek-BGK equation is a kinetic model for alignment of particles moving with constant speed between stochastic reorientation events with sampling from a von Mises distribution. The spatially homogeneous model shows a steady state bifurcation with exchange of stability. The main result of this work is an extension of the bifurcation result to the spatially inhomogeneous problem under the additional assumption of a sufficiently large Knudsen number. The mathematical core is the proof of linearized stability, which employs a new hypocoercivity approach based on Laplace-Fourier transformation. The bifurcation result includes global existence of smooth solutions for close-to-equilibrium initial data. For large data smooth solutions might blow up in finite time whereas weak solutions with bounded Boltzmann entropy are shown to exist globally. |
| title | Stability of equilibria of the spatially inhomogeneous Vicsek-BGK equation across a bifurcation |
| topic | Analysis of PDEs Adaptation and Self-Organizing Systems 35Q92, 35B32 |
| url | https://arxiv.org/abs/2404.00347 |