Averaging regularity results for PDEs under transversality assumptions

Abstract

Let u = u ( x , y ) and f = f ( x , y ) be two functions related by a PDE P ( x , y , Ox, Dy)u = f; the regularity of the y-average s u ( x , .)dy as a function of x is investigated knowing that of u and f.Our method consists in reducing P to a microlocal normal form under a natural transversality assumption. The 2-microlocal regularity of u is also determined knowing that off. These results are then applied to a homogenization problem. This article generalizes the results of [ 161 and [ 121 on velocity averaging and homogenization for kinetic equations.

Stationary Solutions to the Boltzmann Equation in the Hydrodynamic Limit

“…x,v . In the absence of the external field and the boundary, ε 2 Φ ≡ 0 and Ω = R 3 , such gain of integrability is well-known from the Averaging Lemma [24,22] and the sharp Sobolev embedding H 1/2 ⊂ L 3 (See also the case for a convex bounded domain with ε 2 Φ ≡ 0 in [24]). We need to extend this estimate properly to case of the bounded domain Ω in the presence of the external field ε 2 Φ = 0.…”

Homogenization and Diffusion Asymptotics of the Linear Boltzmann Equation

“…The question of diffusion approximation of kinetic processes (limit λ/L 1 and /L ∼ 1) has motivated a lot of works, with various fields of applications: neutron transport by Larsen and Keller [35], Rosseland approximation for radiative transfer problems by Bardos et al [7] and Bardos et al [8], discrete-velocity models by Lions and Toscani [37], semiconductors Boltzmann equations by Poupaud [40], semiconductors Boltzmann equations with Pauli exclusion principle by Golse and Poupaud [26] and Goudon and Mellet [28], general linear equation without detailed balance relation by Degond et al [17], chemotaxis phenomena by Chalub et al [14]... On the other hand, we can search for effective transport coefficients (limit /L 1 and λ/L ∼ 1), as in Dumas and Golse [19] and Gérard and Golse [23]. In our case, the ordering assumes that these length scales have the same order.…”

On propagation of higher space regularity for nonlinear Vlasov equations

“…Uj (H j )(t, x), ¶ Actually this can be embedded in a more general framework, see in particular [24,23,25].…”