A Hamilton-Jacobi approach to nonlocal kinetic equations
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916110412021760 |
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| author | Loy, Nadia Perthame, Benoit |
| author_facet | Loy, Nadia Perthame, Benoit |
| contents | Highly concentrated patterns have been observed in a spatially heterogeneous, nonlocal, model of BGK type implementing a velocity-jump process.
We study both a linear and a nonlinear case and describe the concentration profile. In particular, we analyse a hyperbolic (or high frequency) regime that can be interpreted both as a local (microscopic) or as a nonlocal (macroscopic) rescaling. We consider a Hopf-Cole transform and derive a Hamilton-Jacobi equation. The concentrations are then explained as a consequence of the stationary points of the Hamiltonian that is spatially heterogeneous like the velocity-jump process. After revising the classical hydrodynamic limits for the aggregate quantities and the eikonal equation that can be derived from those with a Hopf-Cole transform, we find that the Hamilton-Jacobi equation is a second order approximation of the eikonal equation in the limit of small diffusivity. For nonlinear turning kernels, the Hopf-Cole transform allows to study the stability of the possible homogeneous configurations and of patterns and the results of a linear stability analysis previously obtained are found and extended to a nonlinear regime. In particular, it is shown that instability (pattern formation) occurs when the Hamiltonian is convex-concave. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_17176 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Hamilton-Jacobi approach to nonlocal kinetic equations Loy, Nadia Perthame, Benoit Mathematical Physics Adaptation and Self-Organizing Systems Highly concentrated patterns have been observed in a spatially heterogeneous, nonlocal, model of BGK type implementing a velocity-jump process. We study both a linear and a nonlinear case and describe the concentration profile. In particular, we analyse a hyperbolic (or high frequency) regime that can be interpreted both as a local (microscopic) or as a nonlocal (macroscopic) rescaling. We consider a Hopf-Cole transform and derive a Hamilton-Jacobi equation. The concentrations are then explained as a consequence of the stationary points of the Hamiltonian that is spatially heterogeneous like the velocity-jump process. After revising the classical hydrodynamic limits for the aggregate quantities and the eikonal equation that can be derived from those with a Hopf-Cole transform, we find that the Hamilton-Jacobi equation is a second order approximation of the eikonal equation in the limit of small diffusivity. For nonlinear turning kernels, the Hopf-Cole transform allows to study the stability of the possible homogeneous configurations and of patterns and the results of a linear stability analysis previously obtained are found and extended to a nonlinear regime. In particular, it is shown that instability (pattern formation) occurs when the Hamiltonian is convex-concave. |
| title | A Hamilton-Jacobi approach to nonlocal kinetic equations |
| topic | Mathematical Physics Adaptation and Self-Organizing Systems |
| url | https://arxiv.org/abs/2401.17176 |