Semi-local behaviour of non-local hypoelliptic equations: Boltzmann
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915104675594240 |
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| author | Loher, Amélie |
| author_facet | Loher, Amélie |
| contents | The purpose of this note is to demonstrate the announced result in [Loher, The Strong Harnack inequality for the Boltzmann equation, Séminaire Laurent Schwartz proceeding] by filling the gap in the proof sketch. We prove the semi-local Strong Harnack inequality for the Boltzmann equation for moderately soft potentials without cutoff assumption. The non-local operator in the Boltzmann equation is in non-divergence form, and thus the method developed in [arXiv:2404.05612] does not apply. However, we exploit that the Boltzmann equation is on average in divergence form, and we show that the non-divergent part of the collision operator is of lower order in a suitable sense, which proves to be sufficient to deduce the Strong Harnack inequality. Consequentially, we derive upper and lower bounds on the fundamental solution of the linearised Boltzmann equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02903 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Semi-local behaviour of non-local hypoelliptic equations: Boltzmann Loher, Amélie Analysis of PDEs 45K05, 35H10, 35Q20, 35B45, 35B65 The purpose of this note is to demonstrate the announced result in [Loher, The Strong Harnack inequality for the Boltzmann equation, Séminaire Laurent Schwartz proceeding] by filling the gap in the proof sketch. We prove the semi-local Strong Harnack inequality for the Boltzmann equation for moderately soft potentials without cutoff assumption. The non-local operator in the Boltzmann equation is in non-divergence form, and thus the method developed in [arXiv:2404.05612] does not apply. However, we exploit that the Boltzmann equation is on average in divergence form, and we show that the non-divergent part of the collision operator is of lower order in a suitable sense, which proves to be sufficient to deduce the Strong Harnack inequality. Consequentially, we derive upper and lower bounds on the fundamental solution of the linearised Boltzmann equation. |
| title | Semi-local behaviour of non-local hypoelliptic equations: Boltzmann |
| topic | Analysis of PDEs 45K05, 35H10, 35Q20, 35B45, 35B65 |
| url | https://arxiv.org/abs/2409.02903 |