Abstract
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.
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References
C. Bardos, F. Golse & C.D. Levermore. Fluid Dynamic Limits of Kinetic Equations II: Convergence Proofs for the Boltzmann Equation, Comm. Pure Appl. Math., 46(1993), 667–753.
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.
R.J. DiPerna & P.L. Lions. On the Cauchy Problem for the Boltzmann Equation: Global Existence and Weak Stability Results. Annals of Math., 130 (1990), 321–366.
F. Golse & C.D. Levermore. The Stokes-Fourier and Acoustic Limits for the Boltzmann Equation, Comm. Pure Appl. Math.
F. Golse & L. Saint-Raymond. The Navier-Stokes Limit of the Boltzmann Equation for Bounded Collision Kernels, Invent. Math., 55 (2004), 81–161.
H. Grad. Asymptotic theory of the Boltzmann equation II, Proc. 3rd Internat. Sympos., Palais de l’Unesco 1 (1963), Paris.
C.D. Levermore & N. Masmoudi. From the Boltzmann Equation to an Incompressible Navier-Stokes-Fourier System. Preprint.
P.L. Lions & N. Masmoudi. From Boltzmann Equations to the Stokes and Euler Equations II, Arch. Ration. Mech. Anal., 158 (2001), 195–211.
P.L. Lions & N. Masmoudi. From Boltzmann Equations to Navier-Stokes Equations I, Arch. Ration. Mech. Anal., 158 (2001), 173–193.
S. Mischler. On weak-weak convergences and applications to the initial boundary value problem for kinetic equations. Preprint.
N. Masmoudi & L. Saint-Raymond. From the Boltzmann equation to the Stokes-Fourier system in a bounded domain, Comm. Pure Appl. Math., 56(2003), 1263–1293.
L. Saint-Raymond. Convergence of solutions to the Boltzmann equation in the incompressible Euler limit, Arch. Ration. Mech. Anal., 166 (2003), 47–80.
L. Saint-Raymond. From the BGK model to the Navier-Stokes equations, Ann. Sci. École Norm. Sup., 36 (2003), 271–317.
H.T. Yau. Relative entropy and hydrodynamics of Ginzburg-Landau models. Letters in Math. Phys., 22 (1991), 63–80.
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© 2006 Birkhäuser Verlag Basel/Switzerland
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Saint-Raymond, L. (2006). Some Recent Results about the Sixth Problem of Hilbert. In: Calgaro, C., Coulombel, JF., Goudon, T. (eds) Analysis and Simulation of Fluid Dynamics. Advances in Mathematical Fluid Mechanics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7742-7_11
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DOI: https://doi.org/10.1007/978-3-7643-7742-7_11
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7741-0
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