Averaging Lemmas and Dispersion Estimates for kinetic equations
نویسنده
چکیده
Averaging lemmas consist in a regularizing effect on the average of the solution to a linear kinetic equation. Some of the main results are reviewed and their proofs presented in as self contained a way as possible. The use of kinetic formulations for the well posedness of scalar conservation laws is eventually explained as an example of application.
منابع مشابه
A New Approach to Velocity Averaging Lemmas in Besov Spaces
We develop a new approach to velocity averaging lemmas based on the dispersive properties of the kinetic transport operator. This method yields unprecedented sharp results, which display, in some cases, a gain of one full derivative. Moreover, the study of dispersion allows to treat the case of LxL p v integrability with r ≤ p. We also establish results on the control of concentrations in the d...
متن کاملHypoelliptic estimates in radiative transfer
We derive the hypoelliptic estimates for a kinetic equation of the form ∂tf + k · ∇xf = (−∆d)h, for (t, x, k) ∈ R× R × S, where d ≥ 1, β > 0, Sd is the unit sphere in Rd+1 and ∆d is the Laplace-Beltrami operator on Sd. Such equations arise in the modeling of high frequency waves in random media with long-range correlations. Assuming some (fractional) Sobolev regularity in the momentum variable ...
متن کاملRegularity in kinetic formulations via averaging lemmas
We present a new class of averaging lemmas directly motivated by the question of regularity for different nonlinear equations or variational problems which admit a kinetic formulation. In particular they improve the known regularity for systems like γ = 3 in isentropic gas dynamics or in some variational problems arising in thin micromagnetic films. They also allow to obtain directly the best k...
متن کاملOn Strong Local Alignment in the Kinetic Cucker-smale Model
In the recent papers [4, 5] the authors study the existence of weak solutions and the hydrodynamic limit of kinetic flocking equations with strong local alignment. The introduction of a strong local alignment term to model flocking behavior was formally motivated in these papers as a limiting case of an alignment term proposed by Motsch and Tadmor [6]. In this paper, we rigorously justify this ...
متن کاملKinetic Semidiscretization of Scalar Conservation Laws and Convergence by Using Averaging Lemmas∗
We consider a time discrete kinetic scheme (known as “transport collapse method”) for the inviscid Burgers equation ∂tu+ ∂x u 2 = 0. We prove the convergence of the scheme by using averaging lemmas without bounded variation estimate. Then, the extension of this result to the kinetic model of Brenier and Corrias is discussed.
متن کامل