Régularité et compacité pour des noyaux de collision de Boltzmann sans troncature angulaire

doi:10.1016/S0764-4442(97)82709-7

Comptes Rendus de l'Académie des Sciences - Series I - Mathematics

Volume 326, Issue 1, January 1998, Pages 37-41

Comptes Rendus de l'...

https://doi.org/10.1016/S0764-4442(97)82709-7 Get rights and content

Résumé

Nous présentons une preuve simple de phénomènes de régularité et de compacité induits par des noyaux de collision de Boltzmann singuliers.

Abstract

We present a simple proof of regularity and compactness phenomena induced by singular Boltzmann collision kernels.

Références bibliographiques (13)

R. Alexandre: preprint, 1997 et communication...
H. Andreasson: preprint,...
F. Bouchut, L. Desvillettes: A proof of the smoothing properties of the positive part of Boltzmann's kernel....
C. Cercignani
The Boltzmann equation and its applications
(1988)
S. Chapman et al.
The mathematical theory of nonuniform gases
(1952)
L. Desvillettes
About the regularity properties of the noncutoff Kac equation
Comm. Math. Phys.
(1995)

There are more references available in the full text version of this article.

Cited by (55)

Propagation of Gevrey regularity for solution of non-cutoff Boltzmann equation
2022, Nonlinear Analysis: Real World Applications
Citation Excerpt :
Assuming the existence of the smooth solution, we state now the main result of the paper as follows: For the developments of regularity to non-cutoff Boltzmann equation, one can refer to [1,4–6,19,22–35] for instance. Before our proof, we first give a Proposition which will be proved later.
Boltzmann to Landau from the gradient flow perspective
2022, Nonlinear Analysis, Theory, Methods and Applications
We revisit the grazing collision limit connecting the Boltzmann equation to the Landau(-Fokker–Planck) equation from their recent reinterpretations as gradient flows. Our results are in the same spirit as the $Γ$ -convergence of gradient flows technique introduced by Sandier and Serfaty [39]; Serfaty [41]. In this setting, the grazing collision limit reduces to showing the lower semi-continuous convergence of the Boltzmann entropy-dissipation to the Landau entropy-dissipation.
Entropy dissipation estimates for the Landau equation in the Coulomb case and applications
2015, Journal of Functional Analysis
Citation Excerpt :
Note nevertheless that (in the spatially homogeneous as well as in the spatially inhomogeneous context), it is possible (in the Coulomb case) to build a theory of local (in time) solutions, or of global solutions with small initial data, cf. [5,15,21,2]. This result is related to the estimates on the non-cutoff Boltzmann operator proven in [3] (cf. also the earlier works [23,29,1]), which belong to the class of entropy dissipation estimates. This means that the entropy dissipation functional related to a PDE is bounded below by a functional controlling somehow the smoothness of the function (appearing in the entropy dissipation functional).
We present in this paper an estimate which bounds from below the entropy dissipation $D (f)$ of the Landau operator with Coulomb interaction by a weighted $H^{1}$ norm of the square root of f. As a consequence, we get a weighted $L_{t}^{1} (L_{v}^{3})$ estimate for the solutions of the spatially homogeneous Landau equation with Coulomb interaction, and the propagation of $L^{1}$ moments of any order for this equation. We also present an application of our estimate to the Landau equation with (moderately) soft potentials, providing thus a new proof of some recent results of [30].
The Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential
2012, Journal of Functional Analysis
It is known that the singularity in the non-cutoff cross-section of the Boltzmann equation leads to the gain of regularity and a possible gain of weight in the velocity variable. By defining and analyzing a non-isotropic norm which precisely captures the dissipation in the linearized collision operator, we first give a new and precise coercivity estimate for the non-cutoff Boltzmann equation for general physical cross-sections. Then the Cauchy problem for the Boltzmann equation is considered in the framework of small perturbation of an equilibrium state. In this part, for the soft potential case in the sense that there is no positive power gain of weight in the coercivity estimate on the linearized operator, we derive some new functional estimates on the nonlinear collision operator. Together with the coercivity estimates, we prove the global existence of classical solutions for the Boltzmann equation in weighted Sobolev spaces.
Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production
2011, Advances in Mathematics
This article provides sharp constructive upper and lower bound estimates for the Boltzmann collision operator with the full range of physical non-cut-off collision kernels ( $γ > - n$ and $s \in (0, 1)$ ) in the trilinear $L^{2} (R^{n})$ energy $〈 Q (g, f), f 〉$ . These new estimates prove that, for a very general class of $g (v)$ , the global diffusive behavior (on f) in the energy space is that of the geometric fractional derivative semi-norm identified in the linearized context in our earlier works (Gressman and Strain, 2010 [15], 2011 [16]). We further prove new global entropy production estimates with the same anisotropic semi-norm. This resolves the longstanding, widespread heuristic conjecture about the sharp diffusive nature of the non-cut-off Boltzmann collision operator in the energy space $L^{2} (R^{n})$ .
The smoothing effect in sharp Gevrey space for the spatially homogeneous non-cutoff Boltzmann equations with a hard potential
2024, Acta Mathematica Scientia

View all citing articles on Scopus

View full text

Régularité et compacité pour des noyaux de collision de Boltzmann sans troncature angulaireRegularity and compactness for Boltzmann collision kernels without angular cut-off

Résumé

Abstract

The Boltzmann equation and its applications

The mathematical theory of nonuniform gases

About the regularity properties of the noncutoff Kac equation

Comm. Math. Phys.