Some Recent Results about the Sixth Problem of Hilbert
The sixth problem proposed by Hilbert, in the occasion of the International Congress of Mathematicians held in Paris in 1900, asks for a global understanding of the gas dynamics. For a perfect gas, the kinetic equation of Boltzmann provides a suitable model of evolution for the statistical distribution of particles. Hydrodynamic models are obtained as first approximations when collisions are frequent. In incompressible regime, rigorous convergence results are now established by describing precisely the corrections to the hydrodynamic approximation, namely physical phenomena such as relaxation or oscillations on small spatio-temporal scales, and checking that they do not disturb the mean motion.
KeywordsBoltzmann Equation Local Thermodynamic Equilibrium Macroscopic Parameter Incompressible Euler Equation Microscopic Interaction
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- F. Bouchut, F. Golse & M. Pulvirenti. Kinetic Equations and Asymptotic Theory, B. Perthame and L. Desvillettes eds., Series in Applied Mathematics, 4 (2000), Gauthier-Villars, Paris.Google Scholar
- F. Golse & C.D. Levermore. The Stokes-Fourier and Acoustic Limits for the Boltzmann Equation, Comm. Pure Appl. Math. Google Scholar
- H. Grad. Asymptotic theory of the Boltzmann equation II, Proc. 3rd Internat. Sympos., Palais de l’Unesco 1 (1963), Paris.Google Scholar
- C.D. Levermore & N. Masmoudi. From the Boltzmann Equation to an Incompressible Navier-Stokes-Fourier System. Preprint.Google Scholar
- S. Mischler. On weak-weak convergences and applications to the initial boundary value problem for kinetic equations. Preprint.Google Scholar