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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.04069 |
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| _version_ | 1866910560276185088 |
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| author | Alonso, R. Bagland, V. Cañizo, J. A. Lods, B. Throm, S. |
| author_facet | Alonso, R. Bagland, V. Cañizo, J. A. Lods, B. Throm, S. |
| contents | We prove the stability of $L^{1}$ self-similar profiles under the hard-to-Maxwell potential limit
for the one-dimensional inelastic Boltzmann equation with moderately hard potentials which, in
turn, leads to the uniqueness of such profiles for hard potentials collision kernels of the form $|\cdot|^γ$
with $γ>0$ sufficiently small (explicitly quantified). Our result provides the first
uniqueness statement for self-similar profiles of inelastic
Boltzmann models allowing for strong inelasticity besides the
explicitly solvable case of Maxwell interactions (corresponding to
$γ=0$). Our approach relies on a perturbation argument from the
corresponding Maxwell model and a careful study of the
associated linearized operator recently derived in the companion paper \cite{maxwel}. The results can be seen as a
first step towards a complete proof, in the one-dimensional setting, of
a conjecture in \cite{ernst} regarding the determination of the
long-time behaviour of solutions to inelastic Boltzmann equation, at least, in a regime of moderately hard potentials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_04069 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | One-dimensional inelastic Boltzmann equation: Stability and uniqueness of self-similar $L^{1}$-profiles for moderately hard potentials Alonso, R. Bagland, V. Cañizo, J. A. Lods, B. Throm, S. Analysis of PDEs We prove the stability of $L^{1}$ self-similar profiles under the hard-to-Maxwell potential limit for the one-dimensional inelastic Boltzmann equation with moderately hard potentials which, in turn, leads to the uniqueness of such profiles for hard potentials collision kernels of the form $|\cdot|^γ$ with $γ>0$ sufficiently small (explicitly quantified). Our result provides the first uniqueness statement for self-similar profiles of inelastic Boltzmann models allowing for strong inelasticity besides the explicitly solvable case of Maxwell interactions (corresponding to $γ=0$). Our approach relies on a perturbation argument from the corresponding Maxwell model and a careful study of the associated linearized operator recently derived in the companion paper \cite{maxwel}. The results can be seen as a first step towards a complete proof, in the one-dimensional setting, of a conjecture in \cite{ernst} regarding the determination of the long-time behaviour of solutions to inelastic Boltzmann equation, at least, in a regime of moderately hard potentials. |
| title | One-dimensional inelastic Boltzmann equation: Stability and uniqueness of self-similar $L^{1}$-profiles for moderately hard potentials |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2408.04069 |