Abstract
The connection between kinetic theory and the macroscopic equations of fluid dynamics is described. In particular, our results concerning the incompressible Navier-Stokes equations are based on a formal derivation in which limiting moments are carefully balanced rather than on a classical expansion such as those of Hilbert or Chapman-Enskog. The moment formalism shows that the limit leading to the incompressible Navier-Stokes equations, like that leading to the compressible Euler equations, is a natural one in kinetic theory and is contrasted with the systematics leading to the compressible Navier-Stokes equations. Some indications of the validity of these limits are given. More specifically, the connection between the DiPerna-Lions renormalized solution of the classical Boltzmann equation and the Leray solution of the Navier-Stokes equations is discussed.
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References
- 1.
C. Bardos, F. Golse, and D. Levermore, Sur les limites asymptotiques de la théorie cinétique conduisant á la dynamique des fluides incompressibles,Comptes Rendus Acad. Sci. 309-I:727–732 (1989).
- 2.
C. Bardos, F. Golse, and D. Levermore, Fluid dynamic limits of kinetic equations II: Convergence proofs for the Boltzmann equation,Ann. Math., submitted.
- 3.
B. Bayly, D. Levermore, and T. Passot, Density variations in weakly compressible flow,Phys. Fluids, submitted.
- 4.
R. Caflisch, The fluid dynamical limit of the nonlinear Boltzmann equation,Commun. Pure Appl. Math. 33:651–666 (1980).
- 5.
C. Cercignani,The Boltzmann Equation and Its Applications (Springer-Verlag, 1988).
- 6.
S. Chapman and T. Cowling,The Mathematical Theory of Nonuniform Gases (Cambridge University Press, 1951).
- 7.
A. De Masi, R. Esposito, and J. L. Lebowitz, Incompressible Navier-Stokes and Euler limits of the Boltzmann equation,Commun. Pure Appl. Math. 42:1189–1214 (1989).
- 8.
R. J. DiPerna and P.-L. Lions, On the Cauchy problem for the Boltzmann equation: Global existence and weak stability results,Ann. Math. 130:321–366 (1989).
- 9.
F. Golse, P.-L. Lions, B. Perthame, and R. Sentis, Regularity of the moments of the solution of a transport equation,J. Funct. Anal. 76:110–125 (1988).
- 10.
S. Kawashima, A. Matsumura, and T. Nishida, On the fluid dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation,Commun. Math. Phys. 70:97–124 (1979).
- 11.
J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace,Acta Math. 63:193–248 (1934).
- 12.
L. D. Landau and E. M. Lifshitz,Fluid Mechanics (Addison-Wesley).
- 13.
T. Nishida, Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the incompressible Euler equation,Commun. Math. Phys. 61:119–148 (1978).
- 14.
T. Sideris, Formation of singularities in three dimensional compressible fluids,Commun. Math. Phys. 101:475–485 (1985).
- 15.
Y. Sone, Asymptotic theory of flow of a rarefied gas over a smooth boundary I, inRarefied Gas Dynamics, Trilling and Wachman, eds. (Academic Press, New York, 1969), pp. 243–253.
- 16.
Y. Sone and K. Aoki, Steady gas flows past bodies at small Knudsen numbers-Boltzmann and hydrodynamic systems,Trans. Theory Stat Phys. 16:189–199 (1987).
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This paper is dedicated to Joel Lebowitz on his 60th-birthday.
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Bardos, C., Golse, F. & Levermore, D. Fluid dynamic limits of kinetic equations. I. Formal derivations. J Stat Phys 63, 323–344 (1991). https://doi.org/10.1007/BF01026608
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Key words
- Boltzmann equation
- Chapman-Enskog expansion
- incompressible Navier stokes equation
- renormalized and weak solutions