Angular Cutoff Assumption Research Articles

Gevrey smoothing effect of solutions for spatially homogeneous nonlinear Boltzmann equation without angular cutoff

There have been extensive studies on the regularizing effect of solutions to the Boltzmann equation without angular cutoff assumption, for both spatially homogeneous and inhomogeneous cases, by noticing the fact that non cutoff Boltzmann collision operator behaves like the fractional power of the Laplace operator. As a further study on the problem in the spatially homogeneous situation, in this paper, we consider the Gevrey regularity of C ∞ solutions with the Maxwellian decay to the Cauchy problem of spatially homogeneous Boltzmann equation for modified hard potentials, by using analytic techniques developed in Alexandre et al. (J Funct Anal 255:2013–2066, 2008; Arch Ration Mech Anal, doi: 10.1007/s00205-010-0290-1 , 2010), Huo et al. (Kinet Relat Models 1:453–489, 2008) and Morimoto et al. (Discrete Contin Dyn Syst Ser A 24:187–212, 2009).

Regularizing Effect and Local Existence for the Non-Cutoff Boltzmann Equation

The Boltzmann equation without Grad’s angular cutoff assumption is believed to have a regularizing effect on the solutions because of the non-integrable angular singularity of the cross-section. However, even though this has been justified satisfactorily for the spatially homogeneous Boltzmann equation, it is still basically unsolved for the spatially inhomogeneous Boltzmann equation. In this paper, by sharpening the coercivity and upper bound estimates for the collision operator, establishing the hypo-ellipticity of the Boltzmann operator based on a generalized version of the uncertainty principle, and analyzing the commutators between the collision operator and some weighted pseudo-differential operators, we prove the regularizing effect in all (time, space and velocity) variables on the solutions when some mild regularity is imposed on these solutions. For completeness, we also show that when the initial data has this mild regularity and a Maxwellian type decay in the velocity variable, there exists a unique local solution with the same regularity, so that this solution acquires the C ∞ regularity for any positive time.

Uncertainty principle and kinetic equations

A large number of mathematical studies on the Boltzmann equation are based on the Grad's angular cutoff assumption. However, for particle interaction with inverse power law potentials, the associated cross-sections have a non-integrable singularity corresponding to the grazing collisions. Smoothing properties of solutions are then expected. On the other hand, the uncertainty principle, established by Heisenberg in 1927, has been developed so far in various situations, and it has been applied to the study of the existence and smoothness of solutions to partial differential equations. This paper is the first one to apply this celebrated principle to the study of the singularity in the cross-sections for kinetic equations. Precisely, we will first prove a generalized version of the uncertainty principle and then apply it to justify rigorously the smoothing properties of solutions to some kinetic equations. In particular, we give some estimates on the regularity of solutions in Sobolev spaces w.r.t. all variables for both linearized and nonlinear space inhomogeneous Boltzmann equations without angular cutoff, as well as the linearized space inhomogeneous Landau equation.

PROPAGATION OF SINGULARITIES IN THE SOLUTIONS TO THE BOLTZMANN EQUATION NEAR EQUILIBRIUM

This paper is about the propagation of the singularities in the solutions to the Cauchy problem of the spatially inhomogeneous Boltzmann equation with angular cutoff assumption. It is motivated by the work of Boudin–Desvillettes on the propagation of singularities in solutions near vacuum. It shows that for the solution near a global Maxwellian, singularities in the initial data propagate like the free transportation. Precisely, the solution is the sum of two parts in which one keeps the singularities of the initial data and the other one is regular with locally bounded derivatives of fractional order in some Sobolev space. In addition, the dependence of the regularity on the cross-section is also given.

EXISTENCE AND EXPONENTIAL DECAY OF GLOBAL SOLUTIONS TO THE BOLTZMANN EQUATION NEAR MAXWELLIANS

Under Grad angular cutoff assumption, we first establish the global-in-time classical solutions near Maxwellians to the Boltzmann equation in a periodic box. Furthermore, the exponential decay of such a solution is obtained.

Classical Solutions to the Boltzmann Equation for Molecules with an Angular Cutoff

An important class of collision kernels in the Boltzmann theory are governed by the inverse power law, in which the intermolecular potential between two particles is an inverse power of their distance. Under the Grad angular cutoff assumption, global-in-time classical solutions near Maxwellians are constructed in a periodic box for all soft potentials with −3<γ<0.

On the Boltzmann equation for long‐range interactions

AbstractWe study the Boltzmann equation without Grad's angular cutoff assumption. We introduce a suitable renormalized formulation that allows the cross section to be singular in both the angular and the relative velocity variables. Angular singularities occur as soon as one is interested in long‐range interactions, while singularities in the relative velocity variable occur in the study of soft potentials, in particular, Coulomb interaction. Together with several new estimates, this new formulation enables us to prove existence of weak solutions and to give a proof of a conjecture by Lions (appearance of strong compactness) under general, fully realistic assumptions. © 2001 John Wiley &amp; Sons, Inc.

Entropy Dissipation and Long-Range Interactions

We study Boltzmann's collision operator for long-range interactions, i.e., without Grad's angular cut-off assumption. We establish a functional inequality showing that the entropy dissipation controls smoothness of the distribution function, in a precise sense. Our estimate is optimal, and gives a unified treatment of both the linear and the nonlinear cases. We also give simple and self-contained proofs of several useful results that were scattered in previous works. As an application, we obtain several helpful estimates for the Cauchy problem, and for the Landau approximation in plasma physics.

On a model Boltzmann equation without angular cutoff

A model Boltzmann equation (see formulas (1.1.6) -- (1.1.9) below) without Grad's angular cutoff assumption is considered. One proves: 1) the instantaneous smoothing in both position and velocity variables by the evolution semigroup associated to the Cauchy problem for this model; 2) the derivation of the analogue of the Landau-Fokker-Planck equation in the limit when grazing collisions prevail.

On boltzmann equations and fokker—planck asymptotics: Influence of grazing collisions

In this paper, we are interested in the influence of grazing collisions, with deflection angle near π/2, in the space-homogeneous Boltzmann equation. We consider collision kernels given by inverse-sth-power force laws, and we deal with general initial data with bounded mass, energy, and entropy. First, once a suitable weak formulation is defined, we prove the existence of solutions of the spatially homogeneous Boltzmann equation, without angular cutoff assumption on the collision kernel, fors ≥ 7/3. Next, the convergence of these solutions to solutions of the Landau-Fokker-Planck equation is studied when the collision kernel concentrates around the value π/2. For very soft interactions, 2<s<7/3, the existence of weak solutions is discussed concerning the Boltzmann equation perturbed by a diffusion term

Sur l'équation de boltzmann homogène et sa relation avec l'équation de landau-fokker-planck: influence des collisions rasantes

We prove the existence of weak solutions of the spatially homogeneous Boltzmann equation without angular cut-off assumption for inverse s th power molecules with s ≥ 7/3, and general initial data with bounded mass, kinetic energy and entropy. Next, we show the convergence of these solutions to solutions of the Landau-Fokker-Planck equation when the collision kernel concentrates around the value π/2