Inhomogeneous Boltzmann Equation Research Articles

Global hypoelliptic estimates for fractional order kinetic equation

In this paper we study a class of fractional order kinetic equation, which is a linear model of spatially inhomogeneous Boltzmann equation without angular cutoff. Using the multiplier method introduced by F. Hérau and K. Pravda‐Starov (J. Math. Pures et Appl., 2011), we establish the optimal global hypoelliptic estimates with weights for the linear model operator.

Gevrey regularity for solutions of the non-cutoff Boltzmann equation: The spatially inhomogeneous case

In this paper we consider the non-cutoff Boltzmann equation in the spatially inhomogeneous case. We prove the propagation of Gevrey regularity for the so-called smooth Maxwellian decay solutions to the Cauchy problem of spatially inhomogeneous Boltzmann equation, and obtain Gevrey regularity of order 1/(2s) in the velocity variable v and order 1 in the space variable x. The strategy relies on our recent results for the spatially homogeneous case [T.-F. Zhang and Z. Yin, Gevrey regularity of spatially homogeneous Boltzmann equation without cutoff, J. Differential Equations 253 (4) (2012), 1172–1190. http://dx.doi.org/10.1016/j.jde.2012.04.023]. Rather, we need much more intricate analysis additionally in order to handle with the coupling of the double variables. Combining with the previous result mentioned above, it gives a characterization of the Gevrey regularity of the particular kind of solutions to the non-cutoff Boltzmann.

Inhomogeneous Boltzmann equations: Distance, asymptotics and comparison of the classical and quantum cases

The notion of distance between a global Maxwellian function and an arbitrary solution f (with the same total density ρ at the fixed moment t) of Boltzmann equation is introduced. In this way we essentially generalize the important Kullback–Leibler distance, which was used before. Namely, we generalize it for the spatially inhomogeneous case. An extremal problem to find a solution of the Boltzmann equation, such that dist{M,f} is minimal in the class of solutions with the fixed values of energy and of n moments, is solved. The cases of the classical and quantum (for Fermi and Bose particles) Boltzmann equations are studied and compared. The asymptotics and stability of solutions of the Boltzmann equations are also considered.

Hypoelliptic Estimates for a Linear Model of the Boltzmann Equation Without Angular Cutoff

In this paper, we establish optimal hypoelliptic estimates for a class of kinetic equations, which are simplified linear models for the spatially inhomogeneous Boltzmann equation without angular cutoff.

A note from the Editorial Board

From this year, Professor Seiji Ukai will step down from the managing editorial board of KRM. Seiji Ukai is a world leading expert in the field of mathematical theories of kinetic equations. He is the first one who proved the global existence of solutions to the space inhomogeneous Boltzmann equation in 1974, and he has been making important contributions to this area, including his series of recent works on the Boltzmann equation without angular cutoff. We are deeply indebted to the contributions that Seiji made to the area and the journal and wish him all the best in the future. <br><br> For more information please click the "Full Text" above.

The Boltzmann Equation Without Angular Cutoff in the Whole Space: Qualitative Properties of Solutions

This is a continuation of our series of works for the inhomogeneous Boltzmann equation. We study qualitative properties of classical solutions, precisely, the full regularization in all variables, uniqueness, non-negativity and convergence rate to the equilibrium. Together with the results of Parts I and II about the well posedness of the Cauchy problem around Maxwellian, we conclude this series with a satisfactory mathematical theory for Boltzmann equation without angular cutoff.

A Revision on Classical Solutions to the Cauchy Boltzmann Problem for Soft Potentials

This short note complements the recent paper of the authors (Alonso, Gamba in J. Stat. Phys. 137(5–6):1147–1165, 2009). We revisit the results on propagation of regularity and stability using Lp estimates for the gain and loss collision operators which had the exponent range misstated for the loss operator. We show here the correct range of exponents. We require a Lebesgue’s exponent α>1 in the angular part of the collision kernel in order to obtain finiteness in some constants involved in the regularity and stability estimates. As a consequence the Lp regularity associated to the Cauchy problem of the space inhomogeneous Boltzmann equation holds for a finite range of p≥1 explicitly determined.

Gevrey Hypoellipticity for a Class of Kinetic Equations

In this paper, we study the Gevrey regularity of weak solutions for a class of linear and semi-linear kinetic equations, which are the linear model of spatially inhomogeneous Boltzmann equations without an angular cutoff.

Global well-posedness theory for the spatially inhomogeneous Boltzmann equation without angular cutoff

We present the first global well-posedness result for the Boltzmann equation without angular cutoff in the framework of weighted Sobolev spaces, in a close to equilibrium framework, and for Maxwellian molecules. These solutions become smooth for any positive time. An important ingredient of the proof rests on the introduction of a new norm, encoding both the singularity and the dissipation properties of the linearized collision operator.

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.

Existence of local solutions for the Boltzmann equation without angular cutoff

We consider the spatially inhomogeneous Boltzmann equation without angular cutoff. We prove the existence and uniqueness of local classical solutions to the Cauchy problem, in the function space with Maxwellian type exponential decay with respect to the velocity variable. To cite this article: R. Alexandre et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

UNIFORM Lp-STABILITY ESTIMATE FOR THE SPATIALLY INHOMOGENEOUS BOLTZMANN EQUATION NEAR VACUUM

We present a new uniform Lp-stability theory for the spatially inhomogeneous Boltzmann equation near vacuum via the nonlinear functional approach proposed by the first author. Our stability analysis is based on new nonlinear functionals which are equivalent to the pth power of Lp-distance. The L1-nonlinear functionals play the key role of "modulators" which make the accumulative functional be non-increasing in time t along classical solutions.

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.

Uniform stability estimates for solutions and their gradients to the Boltzmann equation: A unified approach

As to the Cauchy problem for the spatially inhomogeneous Boltzmann equation with cut-off, we prove uniform stability estimates for solutions and their gradients in a unified and elementary way.

Fast numerical solution of Boltzmann equation

AbstractIn this work a new method is proposed for the solution of a spatially inhomogeneous Boltzmann equation. The key idea is to approximate the velocity space with multilinear tensors. This approach allows to solve huge numerical challenges with millions of space discretisations with the complexity of Navies‐Stocks solvers. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Uncertainty principle and regularity for Boltzmann type equations

We give a generalized version of uncertainty principle, and apply it to the study of regularization properties of solutions to kinetic equations. In particular, both linearized and nonlinear space inhomogeneous Boltzmann equations without Grad's cutoff assumption are considered. To cite this article: R. Alexandre et al., C. R. Acad. Sci. Paris, Ser. I 345 (2007).

Interpretation of the TRADE In-Pile source multiplication experiments

Within the framework of the neutronic characterization of the TRIGA RC-1 reactor in support to the TRADE (TRiga Accelerator Driven Experiment) program, the interpretation of the subcriticality level measurements performed in static regime during the TRADE In-Pile experimental program is presented. Different levels of subcriticality have been measured using the MSA (Modified Source Approximated) method by the insertion of a standard fixed radioactive source into different core positions. Starting from a reference configuration, fuel elements were removed: control rods were moved outward as required for the coupling experiments envisioned with the proton accelerator and fission chambers were inserted in order to measure subcritical count rates. A neutron-physics analysis based on the modified formulation of the source multiplication method (MSM) has been carried out, which requires the systematic solution for each experimental configuration of the homogeneous, both in the forward and adjoint forms, and inhomogeneous Boltzmann equations. By means of such a methodology calculated correction factors to be applied to the MSA measured reactivities were produced in order to take into account spatial and energetic effects creating changes in the detector efficiencies and effective source with respect to the calibration configuration. The methodology presented has been tested against a large number of experimental states. The measurements have underlined the sensitivity of the MSA measured reactivities to core geometry changes and control rod perturbations; the efficiency of MSM factors to dramatically correct for this sensitivity is underlined, making of this technique a relevant methodology in view of the incoming US RACE program to be performed in TRIGA reactors.

Impact of Electron–Electron Collisions on the Spatial Electron Relaxation in Non-Isothermal Plasmas

The impact of electron–electron collisions on the spatial relaxation of electrons in the column-anode plasma of a glow discharge, acted upon by a space-independent electric field and initiated by a constant influx at the cathode side of the plasma, is investigated in inert gas plasmas. The investigations are based on a new method for numerically solving the one-dimensional inhomogeneous Boltzmann equation of the electrons including electron–electron interaction in weakly ionized, collision-dominated plasmas. A detailed analysis of the spatial behaviour of the velocity distribution function and relevant macroscopic properties of the electrons is given for various degrees of ionization and electric field strengths. A significant impact of the electron–electron collisions on the relaxation structure and the resultant relaxation length already at relatively low ionization degrees has been found for low to medium electric fields.

On some properties of Markov processes and Monte Carlo methods for inhomogeneous Boltzmann equation

We consider Markov jump processes, Continuous Time Monte Carlo methods [A. I. Khisamutdinov, Simulation statistical modelling of the kinetic equation of rarefied gases. Dokl. Akad. Nauk SSSR (1988) 302, 75–79 (in Russian).] based on these processes, and the inhomogeneous smoothed Boltzmann equation. In the first place, integral forms of a system of master equations for the processes are constructed and general properties of the solution of the system are inspected. Secondly, we investigate unbiased simulation estimators for the computation of linear functionals on the phase density of the average number of particles; and we investigate the asymptotic behaviour of the variance of one of these estimators when the mean initial number density of particles increases. Third, we touch upon the convergence of the phase density to the solution to the Boltzmann equation. The paper considers the general case of a variable number of particles in the system and describes an example of the application of the methods. The results have relation to the known Direct Simulation Monte Carlo methods.