Seok-Bae Yun

Convergence of a Semi-Lagrangian Scheme for the BGK Model of the Boltzmann Equation

Recently, a new class of semi-Lagrangian methods for the BGK model of the Boltzmann equation has been introduced [F. Filbet and G. Russo, Kinet. Relat. Models, 2 (2009), pp. 231–250; G. Russo and P. Santagati, A new class of large time step methods for the BGK models of the Boltzmann equation, arXiv:1103.5247; P. Santagati, High Order Semi-Lagrangian Methods for the BGK Model of the Boltzmann Equation, Ph.D. thesis, University of Catania, Italy, 2007]. These methods work in a satisfactory way either in a rarefied or a fluid regime. Moreover, because of the semi-Lagrangian feature, the stability property is not restricted by the CFL condition. These aspects make them very attractive for practical applications. In this paper, we prove that the discrete solution of the scheme converges in a weighted

Cauchy problem for the Boltzmann-BGK model near a global Maxwellian

L^1

Ellipsoidal BGK Model Near a Global Maxwellian

norm to the unique smooth solution by deriving an explicit error estimate.

UNIFORM L2-STABILITY ESTIMATES FOR THE RELATIVISTIC BOLTZMANN EQUATION

In this paper, we are interested in the Cauchy problem for the Boltzmann-BGK model for a general class of collision frequencies. We prove that the Boltzmann-BGK model linearized around a global Maxwellian admits a unique global smooth solution if the initial perturbation is sufficiently small in a high order energy norm. We also establish an asymptotic decay estimate and uniform L2-stability for nonlinear perturbations.

The Boltzmann Equation Near a Rotational Local Maxwellian

The BGK model has been widely used in place of the Boltzmann equation because of the qualitatively satisfactory results it provides at relatively low computational cost. There is, however, a major drawback to the BGK model: The hydrodynamic limit at the Navier-Stokes level is not correct. One evidence is that the Prandtl number computed using the BGK model does not agree with what is derived from the Boltzmann equation. To overcome this problem, Holway \cite{Holway} introduced the ellipsoidal BGK model where the local Maxwellian is replaced by a non-isotropic Gaussian. In this paper, we prove the existence of classical solutions of the ES-BGK model when the initial data is a small perturbation of the global Maxwellian. The key observation is that the degeneracy of the ellipsoidal BGK model is comparable to that of the original BGK model or the Boltzmann equation in the range

Entropy production estimates for the polyatomic ellipsoidal BGK model

-1/2<\nu<1

Cauchy problem for the ellipsoidal-BGK model of the Boltzmann equation

.

Spatially Homogeneous Boltzmann Equation for Relativistic Particles

In this paper, we derive an a prioriL2-stability estimate for classical solutions to the relativistic Boltzmann equation, when the initial datum is a small perturbation of a global relativistic Maxwellian. For the stability estimate, we use the dissipative property of the linearized collision operator and a Strichartz type estimate for classical solutions. As a direct application of our stability estimates, we establish that classical solutions in Glassey–Strauss and Hsiao–Yus frameworks satisfy a uniform L2-stability estimate.

On a positive decomposition of entropy production functional for the polyatomic BGK model

In rotationally symmetric domains, the Boltzmann equation with specular reflection boundary condition has a special type of equilibrium states called the rotational local Maxwellian which, unlike the uniform Maxwellian, has an additional term related to the angular momentum of the gas. In this paper, we consider the initial boundary value problem of the Boltzmann equation near the rotational local Maxwellian. Based on the L2-L1 framework of [12], we establish the global well-posedness and the convergence toward such equilibrium states.

Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto--Daido model with inertia

Abstract We study the entropy production estimate for the polyatomic ellipsoidal BGK model, which is a relaxation type kinetic model describing the time evolution of polyatomic particle systems. An interesting dichotomy is observed between 0 θ ≤ 1 and θ = 0 : In each case, distinct target Maxwellians should be chosen to estimate the entropy production functional from below by the relative entropy. The time asymptotic equilibrium state toward which the distribution function stabilizes bifurcates accordingly.