Kinetic and Related Models
June 2020 , Volume 13 , Issue 3
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The Boltzmann equation has an angular singularity inherent in the long range interaction between molecules. Angular singularity produces many difficulties in both theoretical and numerical study of Boltzmann equation. As a result, many rely on angular cutoff models to approximate the Boltzmann equation. Cutoff models have no angular singularity and can be solved by existent numerical methods. However, as the singularity goes stronger, the proportion of grazing collision becomes larger, which renders ineffectiveness of the cutoff Boltzmann equation. Based on the theoretical result that the limit of grazing collision is Landau operator, we propose to add a suitably scaled Landau operator to the cutoff equation to form a new approximate equation. This new approximate equation was studied in [
The aim of this paper is to offer an original and comprehensive spectral theoretical approach to the study of convergence to equilibrium, and in particular of the hypocoercivity phenomenon, for contraction semigroups in Hilbert spaces. Our approach rests on a commutation relationship for linear operators known as intertwining, and we utilize this identity to transfer spectral information from a known, reference semigroup
Kaniadakis and Quarati (1994) proposed a Fokker–Planck equation with quadratic drift as a PDE model for the dynamics of bosons in the spatially homogeneous setting. It is an open question whether this equation has solutions exhibiting condensates in finite time. The main analytical challenge lies in the continuation of exploding solutions beyond their first blow-up time while having a linear diffusion term. We present a thoroughly validated time-implicit numerical scheme capable of simulating solutions for arbitrarily long time, and thus enabling a numerical study of the condensation process in the Kaniadakis–Quarati model. We show strong numerical evidence that above the critical mass rotationally symmetric solutions of the Kaniadakis–Quarati model in
We investigate the Vlasov-Poisson equations perturbed by a strong external uniform magnetic field. We study the asymptotic behavior of the solutions, based on averaging techniques. We analyze the case of general initial conditions. By filtering out the oscillations, we are led to a profile. We prove strong convergence results and establish second order estimates.
The Boltzmann equation is a fundamental kinetic equation that describes the dynamics of dilute gas. In this paper we study the local well-posedness of the Boltzmann equation in bounded domain with the Cercignani-Lampis boundary condition, which describes the intermediate reflection law between diffuse reflection and specular reflection via two accommodation coefficients. We prove the local-in-time well-posedness of the equation by establishing an
In this paper, we are concerned with the construction of global-in-time solutions of the Cauchy problem of the Vlasov-Maxwell-Boltzmann system near Maxwellians with strong uniform background magnetic field. The background magnetic field under our consideration can be any given non-zero constant vector rather than vacuum in the previous results available up to now. Our analysis is motivated by the nonlinear energy method developed recently in [
We study local existence and uniqueness of a strong solution to the kinetic thermomechanical Cucker–Smale (in short TCS) model coupled with incompressible Navier–Stokes (NS) equations in the whole space. The coupled system consists of the kinetic TCS equation for particle ensemble and the incompressible NS equations for a fluid via a drag force. For the strong solution, we investigate the blow-up mechanism for the coupled system, and we also study the global existence of a weak solution in the whole space.
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