ISSN:

1937-5093

eISSN:

1937-5077

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## Kinetic & Related Models

March 2011 , Volume 4 , Issue 1

Issue dedicated to Claude-Michel Brauner on the occasion of his 60th birthday

Guest Editors: Jerry L. Bona, Michel Langlais and Alain Miranville

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2011, 4(1): i-v
doi: 10.3934/krm.2011.4.1i

*+*[Abstract](1649)*+*[PDF](2567.3KB)**Abstract:**

On January 7-th, 2010, Carlo Cercignani, Professor of Mathematical Physics at Politecnico di Milano, passed away in Milan, after a long illness.

If one ought to indicate a single man as a reference point for the development of kinetic theory in the last 50 years from a mathematical, physical, historical and, more generally, cultural point of view, there is no doubt that everybody would think of Carlo Cercignani. We have not only lost a very reputed and loved colleague, but also a pivotal figure for all of us working in the field.

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2011, 4(1): 1-16
doi: 10.3934/krm.2011.4.1

*+*[Abstract](2704)*+*[PDF](1140.4KB)**Abstract:**

The Cucker-Smale model for flocking or swarming of birds or insects is generalized to scenarios where a typical bird will be subject to a) a friction force term driving it to fly at optimal speed, b) a repulsive short range force to avoid collisions, c) an attractive "flocking" force computed from the birds seen by each bird inside its vision cone, and d) a "boundary" force which will entice birds to search for and return to the flock if they find themselves at some distance from the flock. We introduce these forces in detail, discuss the required cutoffs and their implications and show that there are natural bounds in velocity space. Well-posedness of the initial value problem is discussed in spaces of measure-valued functions. We conclude with a series of numerical simulations.

2011, 4(1): 17-40
doi: 10.3934/krm.2011.4.17

*+*[Abstract](2802)*+*[PDF](452.2KB)**Abstract:**

We construct bounded classical solutions of the Boltzmann equation in the whole space without specifying any limit behaviors at the spatial infinity and without assuming the smallness condition on initial data. More precisely, we show that if the initial data is non-negative and belongs to a uniformly local Sobolev space in the space variable and a standard Sobolev space with Maxwellian type decay property in the velocity variable, then the Cauchy problem of the Boltzmann equation possesses a unique non-negative local solution in the same function space, both for the cutoff and non-cutoff collision cross section with mild singularity. The known solutions such as solutions on the torus (space periodic solutions) and in the vacuum (solutions vanishing at the spatial infinity), and solutions in the whole space having a limit equilibrium state at the spatial infinity are included in our category.

2011, 4(1): 41-51
doi: 10.3934/krm.2011.4.41

*+*[Abstract](1918)*+*[PDF](347.4KB)**Abstract:**

In this short note we revisit the gain of integrability property of the gain part of the Boltzmann collision operator. This property implies the $W^{l,r}_k$ regularity propagation for solutions of the associated space homogeneous initial value problem. We present a new method to prove the gain of integrability that simplifies the technicalities of previous approaches by avoiding the argument of gain of regularity estimates for the gain collisional integral. In addition our method calculates explicit constants involved in the estimates.

2011, 4(1): 53-85
doi: 10.3934/krm.2011.4.53

*+*[Abstract](1915)*+*[PDF](489.3KB)**Abstract:**

In last years a great interest was brought to molecular transport problems at nanoscales, such as surface diffusion or molecular flows in nano or sub-nano-channels. In a series of papers V. D. Borman, S. Y. Krylov, A. V. Prosyanov and J. J. M. Beenakker proposed to use kinetic theory in order to analyze the mechanisms that determine mobility of molecules in nanoscale channels. This approach proved to be remarkably useful to give new insight on these issues, such as density dependence of the diffusion coefficient. In this paper we revisit these works to derive the kinetic and diffusion models introduced by V. D. Borman, S. Y. Krylov, A. V. Prosyanov and J. J. M. Beenakker by using classical tools of kinetic theory such as scaling and systematic asymptotic analysis. Some results are extended to less restrictive hypothesis.

2011, 4(1): 87-107
doi: 10.3934/krm.2011.4.87

*+*[Abstract](2015)*+*[PDF](418.6KB)**Abstract:**

We investigate the speed of approach to Maxwellian equilibrium for a collisionless gas enclosed in a vessel whose wall are kept at a uniform, constant temperature, assuming diffuse reflection of gas molecules on the vessel wall. We establish lower bounds for potential decay rates assuming uniform $L^p$ bounds on the initial distribution function. We also obtain a decay estimate in the spherically symmetric case. We discuss with particular care the influence of low-speed particles on thermalization by the wall.

2011, 4(1): 109-138
doi: 10.3934/krm.2011.4.109

*+*[Abstract](2263)*+*[PDF](357.8KB)**Abstract:**

We study a rarefied gas, described by the Boltzmann equation, between two coaxial rotating cylinders in the small Knudsen number regime. When the radius of the inner cylinder is suitably sent to infinity, the limiting evolution is expected to converge to a modified Couette flow which keeps memory of the vanishing curvature of the cylinders (

*ghost effect*[18]). In the $1$-d stationary case we prove the existence of a positive isolated $L_2$-solution to the Boltzmann equation and its convergence. This is obtained by means of a truncated bulk-boundary layer expansion which requires the study of a new Milne problem, and an estimate of the remainder based on a generalized spectral inequality.

2011, 4(1): 139-151
doi: 10.3934/krm.2011.4.139

*+*[Abstract](2373)*+*[PDF](344.6KB)**Abstract:**

We propose a new deterministic numerical model, based on the discontinuous Galerkin method, for solving the nonlinear Boltzmann equation for rarefied gases. A set of partial differential equations is derived and analyzed. The new model guarantees the conservation of the mass, momentum and energy for homogeneous solutions. We avoid any stochastic procedures in the treatment of the collision operator of the Boltzmamn equation.

2011, 4(1): 153-167
doi: 10.3934/krm.2011.4.153

*+*[Abstract](2244)*+*[PDF](354.8KB)**Abstract:**

A recently proposed consistent BGK-type approach for chemically reacting gas mixtures is discussed, which accounts for the correct rates of transfer for mass, momentum and energy, and recovers the exact conservation equations and collision equilibria, including mass action law. In particular, the hydrodynamic limit is derived by a Chapman-Enskog procedure, and compared to existing results for the reactive and non-reactive cases.

2011, 4(1): 169-185
doi: 10.3934/krm.2011.4.169

*+*[Abstract](1966)*+*[PDF](397.5KB)**Abstract:**

We study a Maxwell kinetic model of socio-economic behavior introduced in the paper A. V. Bobylev, C. Cercignani and I. M. Gamba, Commun. Math. Phys.,

**291**(2009), 599-644. The model depends on three non-negative parameters $\{\gamma, q ,s\}$ where $0<\gamma\leq 1$ is the control parameter. Two other parameters are fixed by market conditions. Self-similar solution of the corresponding kinetic equation for distribution of wealth is studied in detail for various sets of parameters. In particular, we investigate the efficiency of control. Some exact solutions and numerical examples are presented. Existence and uniqueness of solutions are also discussed.

2011, 4(1): 187-213
doi: 10.3934/krm.2011.4.187

*+*[Abstract](2244)*+*[PDF](1533.8KB)**Abstract:**

We consider an integro-differential model for evolutionary game theory which describes the evolution of a population adopting mixed strategies. Using a reformulation based on the first moments of the solution, we prove some analytical properties of the model and global estimates. The asymptotic behavior and the stability of solutions in the case of two strategies is analyzed in details. Numerical schemes for two and three strategies which are able to capture the correct equilibrium states are also proposed together with several numerical examples.

2011, 4(1): 215-226
doi: 10.3934/krm.2011.4.215

*+*[Abstract](2041)*+*[PDF](355.2KB)**Abstract:**

We consider a positive Vlasov-Helmholtz plasma in interaction with a positive point charge in $\R^2$ and we prove an existence and uniqueness theorem for this system without any assumption on the decay at infinity of the spatial density.

2011, 4(1): 227-258
doi: 10.3934/krm.2011.4.227

*+*[Abstract](3385)*+*[PDF](584.8KB)**Abstract:**

We are concerned with the global well-posedness of a two-phase flow system arising in the modelling of fluid-particle interactions. This system consists of the Vlasov-Fokker-Planck equation for the dispersed phase (particles) coupled to the incompressible Euler equations for a dense phase (fluid) through the friction forcing. Global existence of classical solutions to the Cauchy problem in the whole space is established when initial data is a small smooth perturbation of a constant equilibrium state, and moreover an algebraic rate of convergence of solutions toward equilibrium is obtained under additional conditions on initial data. The proof is based on the macro-micro decomposition and Kawashima's hyperbolic-parabolic dissipation argument. This result is generalized to the periodic case, when particles are in the torus, improving the rate of convergence to exponential.

2011, 4(1): 259-276
doi: 10.3934/krm.2011.4.259

*+*[Abstract](1905)*+*[PDF](370.8KB)**Abstract:**

Steady one-dimensional flame structure is investigated in a binary gas mixture made up by diatomic molecules and atoms, which undergo an irreversible exothermic two--steps reaction, a recombination process followed by inelastic scattering (de-excitation). A kinetic model at the Boltzmann level, accounting for chemical encounters as well as for mechanical collisions, is proposed and its main features are analyzed. In the case of collision dominated regime with slow recombination and fast de-excitation, the model is the starting point for a consistent derivation, via suitable asymptotic expansion of Chapman-Enskog type, of reactive fluid-dynamic Navier-Stokes equations. The resulting set of ordinary differential equations for the smooth steady deflagration profile is investigated in the frame of the qualitative theory of dynamical systems, and numerical results for the flame eigenvalue and for the main macroscopic observables are presented and briefly commented on for illustrative purposes.

2011, 4(1): 277-294
doi: 10.3934/krm.2011.4.277

*+*[Abstract](2567)*+*[PDF](542.6KB)**Abstract:**

Cercignani's conjecture assumes a linear inequality between the entropy and entropy production functionals for Boltzmann's nonlinear integral operator in rarefied gas dynamics. Related to the field of logarithmic Sobolev inequalities and spectral gap inequalities, this issue has been at the core of the renewal of the mathematical theory of convergence to thermodynamical equilibrium for rarefied gases over the past decade. In this review paper, we survey the various positive and negative results which were obtained since the conjecture was proposed in the 1980s.

2011, 4(1): 295-316
doi: 10.3934/krm.2011.4.295

*+*[Abstract](2145)*+*[PDF](1362.5KB)**Abstract:**

We construct a numerical scheme based on the Liouville equation of geometric optics coupled with the Geometric Theory of Diffraction (GTD) to simulate the high frequency linear waves diffracted by a corner. While the reflection boundary conditions are used at the boundary, a diffraction condition, based on the GTD theory, is introduced at the vertex. These conditions are built into the numerical flux for the discretization of the geometrical optics Liouville equation. Numerical experiments are used to verify the validity and accuracy of this new Eulerian numerical method which is able to capture the physical observable of high frequency and diffracted waves without fully resolving the high frequency numerically.

2011, 4(1): 317-331
doi: 10.3934/krm.2011.4.317

*+*[Abstract](2128)*+*[PDF](235.1KB)**Abstract:**

A fourteen moment theory for a granular gas is developed within the framework of the Boltzmann equation where the full contracted moment of fourth order is added to the thirteen moments of mass density, velocity, pressure tensor and heat flux vector. The spatially homogeneous solutions of the fourteen moment theory implied into a time decay of the temperature field which follows closely Haff's law, besides the more accentuated time decays of the pressure deviator, heat flux vector and fourth moment. The requirement that the fourth moment remains constant in time inferred into its identification with the coefficient $a_2$ in the Chapman-Enskog solution of the Boltzmann equation. The laws of Navier-Stokes and Fourier are obtained by restricting to a five field theory and using a method akin to the Maxwellian procedure. The dependence of the heat flux vector on the gradient of the particle number density was obtained thanks to the inclusion of the forth moment. The analysis of the dynamic behavior of small local disturbances from the spatially homogeneous solutions caused by spontaneous internal fluctuations is performed by considering a thirteen field theory and it is shown that for the longitudinal disturbances there exist one hydrodynamic and four kinetic modes, while for the transverse disturbances one hydrodynamic and two kinetic modes are present.

2011, 4(1): 333-344
doi: 10.3934/krm.2011.4.333

*+*[Abstract](2036)*+*[PDF](339.1KB)**Abstract:**

We introduce a $N$-particle system which approximates, in the mean-field limit, the solutions of the Landau equation with Coulomb singularity. This model plays the same role as the Kac's model for the homogeneous Boltzmann equation. We use compactness arguments following [11].

2011, 4(1): 345-359
doi: 10.3934/krm.2011.4.345

*+*[Abstract](24218)*+*[PDF](436.3KB)**Abstract:**

It is often the case in mathematical analysis that solving an open problem can be facilitated by finding a new set of coordinates which may illumniate the known difficulties. In this article, we illustrate how to derive an assortment coordinates in which to represent the relativistic Boltzmann collision operator. We show the equivalence between some known representations [27, 15], and others which seem to be new. One of these representations has been used recently to solve several open problems in [42, 41, 30, 39].

2011, 4(1): 361-384
doi: 10.3934/krm.2011.4.361

*+*[Abstract](2284)*+*[PDF](634.1KB)**Abstract:**

The steady state of a dilute gas enclosed between two infinite parallel plates in relative motion and under the action of a uniform body force parallel to the plates is considered. The Bhatnagar-Gross-Krook model kinetic equation is analytically solved for this Couette-Poiseuille flow to first order in the force and for arbitrary values of the Knudsen number associated with the shear rate. This allows us to investigate the influence of the external force on the non-Newtonian properties of the Couette flow. Moreover, the Couette-Poiseuille flow is analyzed when the shear-rate Knudsen number and the scaled force are of the same order and terms up to second order are retained. In this way, the transition from the bimodal temperature profile characteristic of the pure force-driven Poiseuille flow to the parabolic profile characteristic of the pure Couette flow through several intermediate stages in the Couette-Poiseuille flow are described. A critical comparison with the Navier-Stokes solution of the problem is carried out.

2011, 4(1): 385-399
doi: 10.3934/krm.2011.4.385

*+*[Abstract](2422)*+*[PDF](425.0KB)**Abstract:**

We develop a rigorous formalism for the description of the evolution of observables of quantum systems of particles in the mean-field scaling limit. The corresponding asymptotics of a solution of the initial-value problem of the dual quantum BBGKY hierarchy is constructed. Moreover, links of the evolution of marginal observables and the evolution of quantum states described in terms of a one-particle marginal density operator are established. Such approach gives the alternative description of the kinetic evolution of quantum many-particle systems.

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