Global Well-Posedness in Spatially Critical Besov Space for the Boltzmann Equation

The unique global strong solution in the Chemin–Lerner type space to the Cauchy problem on the Boltzmann equation for hard potentials is constructed in a perturbation framework. Such a solution space is of critical regularity with respect to the spatial variable, and it can capture the intrinsic properties of the Boltzmann equation. For the proof of global well-posedness, we develop some new estimates on the nonlinear collision term through the Littlewood–Paley theory.     

L 2-Stability Theory of the Boltzmann Equation near a Global Maxwellian

We present three a priori L ²-stability estimates for classical solutions to the Boltzmann equation with a cut-off inverse power law potential, when initial datum is a perturbation of a global Maxwellian. We show that L ²-stability estimates of classical solutions depend on Strichartz type estimates of perturbations and the non-positive definiteness of the linearized collision operator. Several well known classical solutions to the Boltzmann equation fit our L ²-stability framework.     

The Acoustic Limit for the Boltzmann Equation

The acoustic equations are the linearization of the compressible Euler equations about a spatially homogeneous fluid state. We first derive them directly from the Boltzmann equation as the formal limit of moment equations for an appropriately scaled family of Boltzmann solutions. We then establish this limit for the Boltzmann equation considered over a periodic spatial domain for bounded collision kernels. Appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations that converge entropically (and hence strongly in L ¹) to a unique limit governed by a solution of the acoustic equations for all time, provided that its initial fluctuations converge entropically to an appropriate limit associated to any given L ² initial data of the acoustic equations. The associated local conservation laws are recovered in the limit. Accepted: October 22, 1999     

Numerical Investigation of the Temperature Macroparameters in a Shock Wave in a Binary Gas Mixture Using the Kinetic Boltzmann Equation

The effect of the Mach number and the concentration and mass ratios on the behavior of the parallel, radial, and total temperatures of the components in a shock wave in a binary gas mixture is studied. The results obtained are compared with the theoretical and experimental results of other investigators.     

Convergence of Discrete-Velocity Schemes for the Boltzmann Equation

In this paper we prove the convergence of two discrete-velocity deterministic schemes for the Boltzmann equation, namely, Buet's scheme and a new finite-volume scheme that we introduce here. We write the discretized equation in the form of a Boltzmann continuous equation in order to be in the framework of the DiPerna-Lions theory of renormalized solutions. In order to prove convergence we have to overcome two difficulties: the convergence of the discretized collision kernel is very weak and the lemma on the compactness of velocity averages can be recovered only asymptotically when the parameter of discretization tends to zero. (Accepted February 6, 1996)     

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<<0.     

Isotropic Hypoellipticity and Trend to Equilibrium for the Fokker-Planck Equation with a High-Degree Potential

We consider the Fokker-Planck equation with a confining or anti-confining potential which behaves at infinity like a possibly high-degree homogeneous function. Hypoellipticity techniques provide the well-posedness of the weak Cauchy problem in both cases as well as instantaneous smoothing and exponential trend to equilibrium. Lower and upper bounds for the rate of convergence to equilibrium are obtained in terms of the lowest positive eigenvalue of the corresponding Witten Laplacian, with detailed applications.     

Nonlinear Stability of Rarefaction Waves for the Boltzmann Equation

It is well known that the Boltzmann equation is related to the Euler and Navier-Stokes equations in the field of gas dynamics. The relation is either for small Knudsen number, or, for dissipative waves in the time-asymptotic sense. In this paper, we show that rarefaction waves for the Boltzmann equation are time-asymptotic stable and tend to the rarefaction waves for the Euler and Navier-Stokes equations. Our main tool is the combination of techniques for viscous conservation laws and the energy method based on micro-macro decomposition of the Boltzmann equation. The expansion nature of the rarefaction waves and the suitable microscopic version of the H-theorem are essential elements of our analysis.     

Global Lp-properties for the spatially homogeneous Boltzmann equation

This paper studies the L ^p-behavior for 1p of solutions of the nonlinear, spatially homogeneous Boltzmann equation for a class of collision kernels including inverse k ^th-power forces with k>5 and angular cut-off. The following topics are treated: differentiability in L ^p together with global boundedness in time for L ^p-moments that exist initially, translation continuity in L ^p uniformly in time, and strong convergence to equilibrium.     

Upper Maxwellian Bounds for the Spatially Homogeneous Boltzmann Equation

For the spatially homogeneous Boltzmann equation with cutoff hard potentials, it is shown that solutions remain bounded from above uniformly in time by a Maxwellian distribution, provided the initial data have a Maxwellian upper bound. The main technique is based on a comparison principle that uses a certain dissipative property of the linear Boltzmann equation. Implications of the technique to propagation of upper Maxwellian bounds in the spatially-inhomogeneous case are discussed.     

The Linear Boltzmann Equation in Column Experiments of Porous Media

Transport in Porous Media - The use of the linear Boltzmann equation is proposed for transport in porous media in a column. By column experiments, we show that the breakthrough curve is reproduced...     

Forced Oscillations of a System,Containing a Snap-Through Truss,Close to Its Equilibrium Position

The nonlinear dynamics of a two-degree-of-freedom mechanical system is considered. This system consists of a linear oscillator under the action of a time-periodic force and a snap-through truss, which acts as an absorber of the forced oscillations of the linear main system. The forced oscillations of the snap-through truss close to its equilibrium position are analyzed by the multiple scales method.     

Regularity Theory for the Spatially Homogeneous Boltzmann Equation with Cut-Off

We develop the regularity theory of the spatially homogeneous Boltzmann equation with cut-off and hard potentials (for instance, hard spheres), by (i) revisiting the L^p theory to obtain constructive bounds, (ii) establishing propagation of smoothness and singularities, (iii) obtaining estimates on the decay of the singularities of the initial datum. Our proofs are based on a detailed study of the regularity of the gain operator. An application to the long-time behavior is presented.     

Global Attractor for Impulsive Reaction-Diffusion Equation

In this paper, we consider a reaction-diffusion equation with nonsmooth nonlinearity whose solutions have impulse effects at fixed moments of time. We show how this object generates a nonautonomous multivalued dynamical system and prove the existence of a compact semiinvariant global attractor in the phase space. __________ Published in Neliniini Kolyvannya, Vol. 8, No. 3, pp. 319–328, July–September, 2005.     

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.     

Boundary Singularity for Thermal Transpiration Problem of the Linearized Boltzmann Equation

We study the boundary singularity of the fluid velocity for the thermal transpiration problem in the kinetic theory. Logarithmic singularity has been demonstrated through the asymptotic and computational analysis. The goal of this paper is to confirm this logarithmic singularity through exact analysis. We use an iterated scheme, with the “gain” part of the collision operator as a source. The iterated scheme is appropriate for large Knudsen numbers considered here and yields an explicit leading term.     

Global Well-Posedness for Compressible Viscoelastic Fluids near Equilibrium

In this paper, we prove local well-posedness for compressible viscoelastic fluids of the Oldroyd model under the assumption that the initial density is bounded away from zero and global well-posedness near equilibrium. The proof of global well-posedness relies on some intrinsic properties of viscoelastic fluids and on a uniform estimate for a linearized hyperbolic–parabolic system with convection terms.