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Kinetic & Related Models

2013 , Volume 6 , Issue 2

Issue dedicated to Michel Frémond on the occasion of his 70th birthday

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Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates
Daniel Balagué , José A. Cañizo and  Pierre Gabriel
2013, 6(2): 219-243 doi: 10.3934/krm.2013.6.219 +[Abstract](97) +[PDF](475.6KB)
We are concerned with the long-time behavior of the growth-frag-mentation equation. We prove fine estimates on the principal eigenfunctions of the growth-fragmentation operator, giving their first-order behavior close to $0$ and $+\infty$. Using these estimates we prove a spectral gap result by following the technique in [1], which implies that solutions decay to the equilibrium exponentially fast. The growth and fragmentation coefficients we consider are quite general, essentially only assumed to behave asymptotically like power laws.
Large deviations for the solution of a Kac-type kinetic equation
Federico Bassetti and  Lucia Ladelli
2013, 6(2): 245-268 doi: 10.3934/krm.2013.6.245 +[Abstract](36) +[PDF](479.7KB)
The aim of this paper is to study large deviations for the self-similar solution of a Kac-type kinetic equation. Under the assumption that the initial condition belongs to the domain of normal attraction of a stable law of index $\alpha < 2$ and under suitable assumptions on the collisional kernel, precise asymptotic behavior of the large deviations probability is given.
Nonlinear stability of a Vlasov equation for magnetic plasmas
Frédérique Charles , Bruno Després , Benoît Perthame and  Rémis Sentis
2013, 6(2): 269-290 doi: 10.3934/krm.2013.6.269 +[Abstract](35) +[PDF](449.9KB)
The mathematical description of laboratory fusion plasmas produced in Tokamaks is still challenging. Complete models for electrons and ions, as Vlasov-Maxwell systems, are computationally too expensive because they take into account all details and scales of magneto-hydrodynamics. In particular, for most of the relevant studies, the mass electron is negligible and the velocity of material waves is much smaller than the speed of light. Therefore it is useful to understand simplified models. Here we propose and study one of those which keeps both the complexity of the Vlasov equation for ions and the Hall effect in Maxwell's equation. Based on energy dissipation, a fundamental physical property, we show that the model is nonlinear stable and consequently prove existence.
The moment guided Monte Carlo method for the Boltzmann equation
Giacomo Dimarco
2013, 6(2): 291-315 doi: 10.3934/krm.2013.6.291 +[Abstract](41) +[PDF](962.4KB)
In this work we propose a generalization of the Moment Guided Monte Carlo method developed in [11]. This approach permits to reduce the variance of the particle methods through a matching with a set of suitable macroscopic moment equations. In order to guarantee that the moment equations provide the correct solutions, they are coupled to the kinetic equation through a non equilibrium term. Here, at the contrary to the previous work in which we considered the simplified BGK operator, we deal with the full Boltzmann operator. Moreover, we introduce an hybrid setting which permits to entirely remove the resolution of the kinetic equation in the limit of infinite number of collisions and to consider only the solution of the compressible Euler equation. This modification additionally reduce the statistical error with respect to our previous work and permits to perform simulations of non equilibrium gases using only a few number of particles. We show at the end of the paper several numerical tests which prove the efficiency and the low level of numerical noise of the method.
Spectral decompositions and $\mathbb{L}^2$-operator norms of toy hypocoercive semi-groups
Sébastien Gadat and  Laurent Miclo
2013, 6(2): 317-372 doi: 10.3934/krm.2013.6.317 +[Abstract](23) +[PDF](668.4KB)
For any $a>0$, consider the hypocoercive generators $y∂_x+a∂_y^2-y∂_y$ and $y∂_x-ax∂_y+∂_y^2-y∂_y$, respectively for $(x,y)\in\mathbb{R}/(2\pi\mathbb{Z})\times\mathbb{R}$ and $(x,y)\in\mathbb{R}\times\mathbb{R}$. The goal of the paper is to obtain exactly the $\mathbb{L}^2(\mu_a)$-operator norms of the corresponding Markov semi-group at any time, where $\mu_a$ is the associated invariant measure. The computations are based on the spectral decomposition of the generator and especially on the scalar products of the eigenvectors. The motivation comes from an attempt to find an alternative approach to classical ones developed to obtain hypocoercive bounds for kinetic models.
Structure of entropies in dissipative multicomponent fluids
Vincent Giovangigli and  Lionel Matuszewski
2013, 6(2): 373-406 doi: 10.3934/krm.2013.6.373 +[Abstract](126) +[PDF](684.7KB)
We investigate the structure of mathematical entropies for dissipative multicomponent fluid models derived from the kinetic theory of gases. The corresponding governing equations notably involve nonideal thermochemistry as well as diffusion fluxes driven by chemical potential gradients and temperature gradients. We obtain the general form of mathematical entropies compatible with the hyperbolic structure of the system of partial differential equations assuming a natural nondegeneracy condition. We next establish that entropies compatible with the hyperbolic-parabolic structure are unique up to an affine transform when they are independent on mass and heat diffusion parameters.
Analytical regularizing effect for the radial and spatially homogeneous Boltzmann equation
Léo Glangetas and  Mohamed Najeme
2013, 6(2): 407-427 doi: 10.3934/krm.2013.6.407 +[Abstract](36) +[PDF](438.0KB)
In this paper, we consider a class of spatially homogeneous Boltzmann equation without angular cutoff. We prove that any radial symmetric weak solution of the Cauchy problem become analytic for positive time.
Collisionless kinetic theory of rolling molecules
Darryl D. Holm , Vakhtang Putkaradze and  Cesare Tronci
2013, 6(2): 429-458 doi: 10.3934/krm.2013.6.429 +[Abstract](61) +[PDF](1051.5KB)
We derive a collisionless kinetic theory for an ensemble of molecules undergoing nonholonomic rolling dynamics. We demonstrate that the existence of nonholonomic constraints leads to problems in generalizing the standard methods of statistical physics. In particular, we show that even though the energy of the system is conserved, and the system is closed in the thermodynamic sense, some fundamental features of statistical physics such as invariant measure do not hold for such nonholonomic systems. Nevertheless, we are able to construct a consistent kinetic theory using Hamilton's variational principle in Lagrangian variables, by regarding the kinetic solution as being concentrated on the constraint distribution. A cold fluid closure for the kinetic system is also presented, along with a particular class of exact solutions of the kinetic equations.

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