"In 1900, at the Second International Congress of Mathematicians, David Hilbert set forth a list of 23 mathematical challenges for the coming century, known as the "Hilbert problems." The sixth of these problems concerns the foundations of classical mechanics. This problem was motivated by the recent work of Ludwig Boltzmann who introduced a description of gases intermediate between atomic models based on Newtonian mechanics, and continuous fluid models.
A natural question is whether these different models give consistent predictions, and Hilbert suggested using methods based on the idea of "boundary crossing" to provide a mathematical answer to this question.
Despite more than a century of effort, the problem raised by Hilbert remains largely unsolved." (Laure Saint-Raymond)
The presentation of May 23, 2023 by Laure Saint-Raymond (Meeting 5 to 7) aims to explain the difficulties encountered in this quest and to show how they can bring an interesting light on physics.
This collection proposes to gather a part of the notes published on this transverse subject in the Comptes Rendus Mathématique, Physique et Mécanique.
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Linear diffusive limit of deterministic systems of hard spheresThierry Bodineau, Isabelle Gallagher, Laure Saint-RaymondComptes Rendus. Mathématique, Tome 352 (2014) no. 5, pp. 411-419 |
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Solutions of the Vlasov–Maxwell–Boltzmann system with long-range interactionsDiogo Arsénio, Laure Saint-RaymondComptes Rendus. Mathématique, Tome 351 (2013) no. 9-10, pp. 357-360 |
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Velocity averaging in L1 for the transport equationFrançois Golse, Laure Saint-RaymondComptes Rendus. Mathématique, Tome 334 (2002) no. 7, pp. 557-562 |
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Pseudodifferential calculus in large dimension and thermodynamic limitsChristophe RoyerComptes Rendus. Mathématique, Tome 336 (2003) no. 5, pp. 413-418 |
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The Boltzmann–Grad limit of the periodic Lorentz gas in two space dimensionsEmanuele Caglioti, François GolseComptes Rendus. Mathématique, Tome 346 (2008) no. 7-8, pp. 477-482 |
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From hard spheres dynamics to the Stokes–Fourier equations: An L2 analysis of the Boltzmann–Grad limitThierry Bodineau, Isabelle Gallagher, Laure Saint-RaymondComptes Rendus. Mathématique, Tome 353 (2015) no. 7, pp. 623-627 |
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Wave packets and the quadratic Monge–Kantorovich distance in quantum mechanicsFrançois Golse, Thierry PaulComptes Rendus. Mathématique, Tome 356 (2018) no. 2, pp. 177-197 |
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Derivation of the Schrödinger–Poisson equation from the quantum N-body problemClaude Bardos, Laszlo Erdös, François Golse, Norbert Mauser, Horng-Tzer YauComptes Rendus. Mathématique, Tome 334 (2002) no. 6, pp. 515-520 |
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A microscopic view of the Fourier lawThierry Bodineau, Isabelle Gallagher, Laure Saint-RaymondComptes Rendus. Physique, Volume 20, Issue 5, July–August 2019, Pages 402-418
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Role of conserved quantities in Fourier's law for diffusive mechanical systemsStefano OllaComptes Rendus. Physique, Volume 20, Issue 5, July–August 2019, Pages 429-441
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The vanishing viscosity as a selection principle for the Euler equations: The case of 3D shear flowClaude Bardos, Edriss S. Titi, Emil WiedemannComptes Rendus. Mathématique, Tome 350 (2012) no. 15-16, pp. 757-760 |
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Observation estimate for kinetic transport equations by diffusion approximationClaude Bardos, Kim Dang PhungComptes Rendus. Mathématique, Tome 355 (2017) no. 6, pp. 640-664 |
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A formalism for the differentiation of conservation lawsClaude Bardos, Olivier PironneauComptes Rendus. Mathématique, Tome 335 (2002) no. 10, pp. 839-845
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Global existence of weak solutions to some micro-macro modelsPierre-Louis Lions, Nader MasmoudiComptes Rendus. Mathématique, Tome 345 (2007) no. 1, pp. 15-20 |
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Stratified radiative transfer for multidimensional fluidsFrançois Golse, Olivier PironneauComptes Rendus. Mécanique, Online first (2022), pp. 1-15 |
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Regularity of solutions for the Boltzmann equation without angular cutoffRadjesvarane Alexandre, Yoshinore Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong YangComptes Rendus. Mathématique, Tome 347 (2009) no. 13-14, pp. 747-752 |
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On steady-state preserving spectral methods for homogeneous Boltzmann equationsFrancis Filbet, Lorenzo Pareschi, Thomas ReyComptes Rendus. Mathématique, Tome 353 (2015) no. 4, pp. 309-314 |
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About Kacʼs program in kinetic theoryStéphane Mischler, Clément MouhotComptes Rendus. Mathématique, Tome 349 (2011) no. 23-24, pp. 1245-1250 |
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Lattice Fluid Dynamics: Thirty-five Years Down the RoadSauro SucciComptes Rendus. Mécanique, Online first (2022), pp. 1-12 |
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A hierarchy of reduced models to approximate Vlasov–Maxwell equations for slow time variationsFranck Assous, Yevgeni FurmanComptes Rendus. Mécanique, Tome 348 (2020) no. 12, pp. 969-981
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High order asymptotic-preserving schemes for the Boltzmann equationGiacomo Dimarco, Lorenzo PareschiComptes Rendus. Mathématique, Tome 350 (2012) no. 9-10, pp. 481-486 |
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An ES–BGK model for gas mixturesStéphane BrullComptes Rendus. Mathématique, Tome 351 (2013) no. 19-20, pp. 775-779 |
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A boundary matching micro/macro decomposition for kinetic equationsMohammed Lemou, Florian MéhatsComptes Rendus. Mathématique, Tome 349 (2011) no. 7-8, pp. 479-484 |
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Kinetic theory and Bose–Einstein condensationColm Connaughton, Yves Pomeau |