Abstract
These three lectures introduce the reader to recent progress on the hydrodynamic limits of the kinetic theory of gases. Lecture 1 outlines the main mathematical results in this direction, and explain in particular how the Euler or Navier-Stokes equations for compressible as well as incompressible fluids, can be derived from the Boltzmann equation. It also presents the notion of renormalized solution of the Boltzmann equation, due to P.-L. Lions and R. DiPerna, together with the mathematical methods used in the proofs of the fluid dynamic limits. Lecture 2 gives a detailed account of the derivation by L. Saint-Raymond of the incompressible Euler equations from the BGK model with constant collision frequency (Saint-Raymond, Bull Sci Math 126:493–506, 2002). Finally, Lecture 3 sketches the main steps in the proof of the incompressible Navier-Stokes limit of the Boltzmann equation, connecting the DiPerna-Lions theory of renormalized solutions of the Boltzmann equation with Leray’s theory of weak solutions of the Navier-Stokes system, following (Golse and Saint-Raymond, J Math Pures Appl 91:508–552, 2009). As is the case of all mathematical results in continuum mechanics, the fluid dynamic limits of the Boltzmann equation involve some basic properties of isotropic tensor fields that are recalled in Appendices 1 and 2.
Keywords
- Fluid Dynamic Limits
- Incompressible Navier-Stokes Limit
- Boltzmann Equation
- Hydrodynamic Limit
- Leray Solution
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Notes
- 1.
If X is a topological space and E is a Banach space, the notation C(X, w − E) designates the set of continuous maps from X to E equipped with its weak topology.
- 2.
This is not completely true, however, since the velocity averaging method is at the heart of the kinetic formulation of hyperbolic conservation laws. Unfortunately, while this approach is rather successful in the case of scalar conservation laws, it seems so far limited to some very special kind of hyperbolic systems: see P.-L. Lions-B. Perthame-E. Tadmor [74], P.-E. Jabin-B. Perthame [61], B. Perthame [83].
- 3.
Let X be a topological space and E a topological vector space. The notation C b (X, E) designates the set of bounded continuous maps from X to E.
- 4.
The Legendre dual f ∗ of a function f: I → R, where I is an interval of R, is defined for all p ∈ R by the formula
$$\displaystyle{{f}^{{\ast}}(p) =\sup _{ x\in I}(\mathit{px} - f(x))\,.}$$ - 5.
Consider the endomorphism of \({({\mathbf{R}}^{N})}^{\otimes 2}\) defined by
$$\displaystyle{u \otimes v\mapsto {(u \otimes v)}^{\sigma } = v \otimes u\,.}$$An element T of \({({\mathbf{R}}^{N})}^{\otimes 2}\) is said to be symmetric if and only if T σ = T.
- 6.
Let G be a subgroup of O N (R) and V be a linear subspace of R N. Assume that G leaves V invariant, i.e. gV ⊂ V for each g ∈ G, and that G acts transitively on the spheres of V centered at 0, i.e. if for each v 1, v 2 ∈ V such that | v 1 | = | v 2 | , there exists g ∈ G satisfying gv 1 = v 2. Then, the only vector v ∈ V such that gv = v for each g ∈ G is v = 0. Indeed, if v ≠ 0, one has | v | = | − v | and therefore there exists g ∈ G such that gv = −v ≠ v.
- 7.
Let G be a subgroup of O N (R) and let V be a linear subspace of R N. Assume that G acts transitively on spheres of V centered at 0. Then the only linear subspaces W of V such that gW ⊂ W for each g ∈ G are {0} and V. Indeed, if W is a linear subspace of V different from either {0} or V, let w ∈ W and z ∈ V ∖ W satisfy | w | = | z | ≠ 0. By the transitivity assumption above, there exists g ∈ G such that gw = z and therefore gW is not included in W.
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Golse, F. (2014). Fluid Dynamic Limits of the Kinetic Theory of Gases. In: Bernardin, C., Gonçalves, P. (eds) From Particle Systems to Partial Differential Equations. Springer Proceedings in Mathematics & Statistics, vol 75. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54271-8_1
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