Hydrodynamic Limit of the Boltzmann Equation toward Generic Riemann Solutions with Shocks
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910250965139456 |
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| author | Choe, Mingi Kang, Moon-jin Kim, Chanwoo |
| author_facet | Choe, Mingi Kang, Moon-jin Kim, Chanwoo |
| contents | We establish the hydrodynamic limit of the one-dimensional Boltzmann equation with hard-sphere collisions toward Riemann solutions of the compressible Euler system. The Riemann solutions covered by our result include generic superpositions of elementary waves: either two shock waves and a contact discontinuity, or a rarefaction wave, a contact discontinuity, and a shock wave. For suitably well-prepared initial data and sufficiently small wave strength, we prove that the corresponding Boltzmann solution exists globally in time and converges, as the Knudsen number vanishes, to the local Maxwellian associated with the Riemann solution in $L^2([0,T]\times\mathbb R_x\times\mathbb R^3_ξ)$ for any $T>0$. The shock locations are modulated by dynamical unknowns, the Shifts, which are obtained as BV functions on $[0,T]$. A distinctive point of our result is that the convergence is proved without removing either the shock layer or the initial layer. In the special case of a single shock, our analysis gives a sharp quantitative description of the kinetic shock layer, up to the dynamically selected Shift. The proof combines the macro-micro decomposition, a kinetic adaptation of the $a$-contraction method for shocks, layer analysis, and compactness arguments for the Shifts. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_24392 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hydrodynamic Limit of the Boltzmann Equation toward Generic Riemann Solutions with Shocks Choe, Mingi Kang, Moon-jin Kim, Chanwoo Analysis of PDEs 76P05, 35Q20, 82B40 We establish the hydrodynamic limit of the one-dimensional Boltzmann equation with hard-sphere collisions toward Riemann solutions of the compressible Euler system. The Riemann solutions covered by our result include generic superpositions of elementary waves: either two shock waves and a contact discontinuity, or a rarefaction wave, a contact discontinuity, and a shock wave. For suitably well-prepared initial data and sufficiently small wave strength, we prove that the corresponding Boltzmann solution exists globally in time and converges, as the Knudsen number vanishes, to the local Maxwellian associated with the Riemann solution in $L^2([0,T]\times\mathbb R_x\times\mathbb R^3_ξ)$ for any $T>0$. The shock locations are modulated by dynamical unknowns, the Shifts, which are obtained as BV functions on $[0,T]$. A distinctive point of our result is that the convergence is proved without removing either the shock layer or the initial layer. In the special case of a single shock, our analysis gives a sharp quantitative description of the kinetic shock layer, up to the dynamically selected Shift. The proof combines the macro-micro decomposition, a kinetic adaptation of the $a$-contraction method for shocks, layer analysis, and compactness arguments for the Shifts. |
| title | Hydrodynamic Limit of the Boltzmann Equation toward Generic Riemann Solutions with Shocks |
| topic | Analysis of PDEs 76P05, 35Q20, 82B40 |
| url | https://arxiv.org/abs/2605.24392 |