Riv.Mat.Univ.Parma (7) 4** (2005)


Hydrodynamic Limits for the Boltzmann Equation

Pages 1-144
Received: 14 September 2005   
Mathematics Subject Classification (2000): 35Q30 -35Q35 - 82C40 - 76P05

Abstract: This article surveys recent mathematical results on the kinetic theory of gases. Specifically, the following topics are discussed in some detail - the global existence theory of R. DiPerna and P.-L. Lions for the Boltzmann equation, and - the derivation of the classical models of fluid mechanics (i.e. the Euler or Navier-Stokes equations) from the Botzmann equation. Among all existing results on these topics, we have chosen to discuss mostly those bearing on solutions that are global in time and for arbitrary initial data satisfying only a priori estimates with intrinsic physical meaning --- typically, bounds on the total mass, energy or entropy. Consequently, the mathematical methods presented here are well adapted to handling limits of sequences of functions with little or no uniform regularity. In particular, we study compactness arguments in Lp spaces implied by controls on derivatives of solutions coming from the partial differential equations satisfied by these solutions.

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