Leif Arkeryd

The Boltzmann equation for weakly inhomogeneous data

We solve the initial value problem associated to the nonlinear Boltzmann equation in the case in which the initial distribution has sufficiently small spatial gradients.

Measure solutions of the steady Boltzmann equation in a slab

It is shown that the steady Boltzmann equation in a slab [0,a] has solutionsx→μx such that the ingoing boundary measuresμ0∣{ξ>0} andμα∣{ξ<0} can be prescribed a priori. The collision kernel is truncated such that particles with smallx-component of the velocity have a reduced collision rate.

Global existence in L1 for the Enskog equation and convergence of the solutions to solutions of the Boltzmann equation

For the Enskog equation in a box an existence theorem is proved for initial data with finite mass, energy, and entropy. Then, by letting the diameter of the molecules go to zero, the weak convergence of solutions of the Enskog equation to solutions of the Boltzmann equation is proved.

On the Enskog equation with large initial data

This paper is concerned with the Enskog equation with large initial data in

On the convergence of solutions of the enskog equation to solutions of the boltzmann equation

L^1

A global existence theorem for the initial-boundary-value problem for the Boltzmann equation when the boundaries are not isothermal

, where the high density factor is constant. As a preliminary step, existence and uniqueness is first studied in full physical space and in a box with periodic boundary conditions under the restriction of bounded velocities, by the use of a priori estimates in the norm

Loeb solutions of the boltzmann equation

\int {(\sup _{0 \leqq t \leqq T} {f(x + tv,v,t)} |)dx\,dv}

On diffuse reflection at the boundary for the Boltzmann equation and related equations

. Global existence and uniqueness for small data and unbounded velocities is an easy consequence of this step. The rest of the paper is devoted to the central topic: global existence, regularity, and uniqueness for large initial data in full physical space for the case of unbounded velocities, provided all v-moments are initially finite. Here the more detailed structure of the collision operator is exploited in the a priori estimates.

On the Milne Problem and the Hydrodynamic Limit for a Steady Boltzmann Equation Model

For the Enskog equation with a symmetrized kernel in a box an existence theorem is proved for initial data with finite mass, energy and entropy. Then by letting the diameter of the molecules go to zero we prove the weak convergence of solutions of the Enskog equation to solutions of the Boltzmann equation.

On the Enskog equation in two space variables

We extend the existence theorem recently proved by Hamdache for the initial-boundary-value problem for the nonlinear Boltzmann equation in a vessel with isothermal boundaries to more general situations including the case when the boundaries are not isothermal. In the latter case a cut-off for large speeds is introduced in the collision term of the Boltzmann equation.