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