On the fractional diffusion for the linear Boltzmann equation with drift and general cross-section
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909535830016000 |
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| author | Dechicha, Dahmane |
| author_facet | Dechicha, Dahmane |
| contents | This paper is devoted to the hydrodynamic limit for the linear Boltzmann equation, in the case of a heavy tail equilibrium and a cross section which depends on the space variable and which degenerates for large velocities, without symmetry assumptions. For an appropriate time scale, a macroscopic equation with an elliptic operator, which is equivalent to the fractional Laplacian, is obtained. This problem has been addressed in [Mellet, Mischler and Mouhot, ARMA, 2011] for a space-independent cross section, using a Fourier-Laplace transform, where a fractional diffusion equation has been obtained, and revisited in [Mellet, Indiana Univ., 2010] for a space-dependent but a bounded cross section, using the moments method by introducing an auxiliary problem. In this work, we will adapt the latter method to generalize both results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_09589 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the fractional diffusion for the linear Boltzmann equation with drift and general cross-section Dechicha, Dahmane Analysis of PDEs 82C40, 35B25, 35F10, 35R11 This paper is devoted to the hydrodynamic limit for the linear Boltzmann equation, in the case of a heavy tail equilibrium and a cross section which depends on the space variable and which degenerates for large velocities, without symmetry assumptions. For an appropriate time scale, a macroscopic equation with an elliptic operator, which is equivalent to the fractional Laplacian, is obtained. This problem has been addressed in [Mellet, Mischler and Mouhot, ARMA, 2011] for a space-independent cross section, using a Fourier-Laplace transform, where a fractional diffusion equation has been obtained, and revisited in [Mellet, Indiana Univ., 2010] for a space-dependent but a bounded cross section, using the moments method by introducing an auxiliary problem. In this work, we will adapt the latter method to generalize both results. |
| title | On the fractional diffusion for the linear Boltzmann equation with drift and general cross-section |
| topic | Analysis of PDEs 82C40, 35B25, 35F10, 35R11 |
| url | https://arxiv.org/abs/2503.09589 |