Comptes Rendus

Putting physics into equations: a problem of mathematics?

"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.

	Linear diffusive limit of deterministic systems of hard spheres Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond Comptes Rendus. Mathématique, Tome 352 (2014) no. 5, pp. 411-419
	Solutions of the Vlasov–Maxwell–Boltzmann system with long-range interactions Diogo Arsénio, Laure Saint-Raymond Comptes Rendus. Mathématique, Tome 351 (2013) no. 9-10, pp. 357-360
	Velocity averaging in L1 for the transport equation François Golse, Laure Saint-Raymond Comptes Rendus. Mathématique, Tome 334 (2002) no. 7, pp. 557-562
	Pseudodifferential calculus in large dimension and thermodynamic limits Christophe Royer Comptes Rendus. Mathématique, Tome 336 (2003) no. 5, pp. 413-418
	The Boltzmann–Grad limit of the periodic Lorentz gas in two space dimensions Emanuele Caglioti, François Golse Comptes Rendus. Mathématique, Tome 346 (2008) no. 7-8, pp. 477-482
	From hard spheres dynamics to the Stokes–Fourier equations: An L2 analysis of the Boltzmann–Grad limit Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond Comptes Rendus. Mathématique, Tome 353 (2015) no. 7, pp. 623-627
	Wave packets and the quadratic Monge–Kantorovich distance in quantum mechanics François Golse, Thierry Paul Comptes Rendus. Mathématique, Tome 356 (2018) no. 2, pp. 177-197
	Derivation of the Schrödinger–Poisson equation from the quantum N-body problem Claude Bardos, Laszlo Erdös, François Golse, Norbert Mauser, Horng-Tzer Yau Comptes Rendus. Mathématique, Tome 334 (2002) no. 6, pp. 515-520
	A microscopic view of the Fourier law Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond Comptes Rendus. Physique, Volume 20, Issue 5, July–August 2019, Pages 402-418
	Role of conserved quantities in Fourier's law for diffusive mechanical systems Stefano Olla Comptes Rendus. Physique, Volume 20, Issue 5, July–August 2019, Pages 429-441
	The vanishing viscosity as a selection principle for the Euler equations: The case of 3D shear flow Claude Bardos, Edriss S. Titi, Emil Wiedemann Comptes Rendus. Mathématique, Tome 350 (2012) no. 15-16, pp. 757-760
	Observation estimate for kinetic transport equations by diffusion approximation Claude Bardos, Kim Dang Phung Comptes Rendus. Mathématique, Tome 355 (2017) no. 6, pp. 640-664
	A formalism for the differentiation of conservation laws Claude Bardos, Olivier Pironneau Comptes Rendus. Mathématique, Tome 335 (2002) no. 10, pp. 839-845
	Global existence of weak solutions to some micro-macro models Pierre-Louis Lions, Nader Masmoudi Comptes Rendus. Mathématique, Tome 345 (2007) no. 1, pp. 15-20
	Stratified radiative transfer for multidimensional fluids François Golse, Olivier Pironneau Comptes Rendus. Mécanique, Online first (2022), pp. 1-15
	Regularity of solutions for the Boltzmann equation without angular cutoff Radjesvarane Alexandre, Yoshinore Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang Comptes Rendus. Mathématique, Tome 347 (2009) no. 13-14, pp. 747-752
	On steady-state preserving spectral methods for homogeneous Boltzmann equations Francis Filbet, Lorenzo Pareschi, Thomas Rey Comptes Rendus. Mathématique, Tome 353 (2015) no. 4, pp. 309-314
	About Kacʼs program in kinetic theory Stéphane Mischler, Clément Mouhot Comptes Rendus. Mathématique, Tome 349 (2011) no. 23-24, pp. 1245-1250
	Lattice Fluid Dynamics: Thirty-five Years Down the Road Sauro Succi Comptes Rendus. Mécanique, Online first (2022), pp. 1-12
	A hierarchy of reduced models to approximate Vlasov–Maxwell equations for slow time variations Franck Assous, Yevgeni Furman Comptes Rendus. Mécanique, Tome 348 (2020) no. 12, pp. 969-981
	High order asymptotic-preserving schemes for the Boltzmann equation Giacomo Dimarco, Lorenzo Pareschi Comptes Rendus. Mathématique, Tome 350 (2012) no. 9-10, pp. 481-486
	An ES–BGK model for gas mixtures Stéphane Brull Comptes Rendus. Mathématique, Tome 351 (2013) no. 19-20, pp. 775-779
	A boundary matching micro/macro decomposition for kinetic equations Mohammed Lemou, Florian Méhats Comptes Rendus. Mathématique, Tome 349 (2011) no. 7-8, pp. 479-484
	Kinetic theory and Bose–Einstein condensation Colm Connaughton, Yves Pomeau Comptes Rendus. Physique, Volume 5, Issue 1, January–February 2004, Pages 91-106