A priori estimates for solutions of a system describing the interaction of gravitationally attracting particles with a self-similar pressure term are proved. The presented theory covers the case of the model with diffusions that obey either Fermi-Dirac statistics or a polytropic one.

The first part reviews some recent ideas and L¹-existence results for non-linear stationary equations of Boltzmann type in a bounded domain in ℝⁿ and far from global Maxwellian equilibrium. That is an area not covered by the DiPerna and P. L. Lions methods for the time-dependent Boltzmann equation from the late 1980-ies. The final part discusses the more classical perturbative case close to global equilibrium and corresponding small mean free path limits of fully non-linear stationary problems....

The author proves the existence of solution of Van Roosbroeck's system of partial differential equations from the theory of semiconductors. His results generalize those of Mock, Gajewski and Seidman.

This paper concerns $l$-velocity model of the general linear time-dependent transport equation. The assumed probability of the collision (scattering, fission) depends only on the angle of the directions of the moving neutron before and after the collision. The weak formulation of the problem is given and a priori estimates are obtained. The construction of an approximate problem by $bad\; hbox$-method is given. In the symmetric hyperbolic system obtained by $bad\; hbox$-method dissipativity and $𝒜$-orthogonality of the relevant...

For general interacting particle systems in the sense of Liggett, it is proven that the class of cylinder functions forms a core for the associated Markov generator. It is argued that this result cannot be concluded by straightforwardly generalizing the standard proof technique that is applied when constructing interacting particle systems from their Markov pregenerators.

We present the semi-conductor Boltzmann equation, which is time-reversible, and indicate that it can be formally derived by considering the large time and small perturbing potential limit in the Von-Neumann equation (time-reversible). We then rigorously compute the corresponding asymptotics in the case of the Von-Neumann equation on the Torus. We show that the limiting equation we obtain does not coincide with the physically realistic model. The former is indeed an equation of Boltzmann type, yet...

Consider the domain $Z_{ϵ}=\{x\in {ℝ}^n;dist(x,ϵ{\mathbb{Z}}^n)>{ϵ}^{\gamma }\}$ and let the free path length be defined as ${\tau }_{ϵ}(x,v)=inf\{t>0;x-tv\in Z_{ϵ}\}.$ In the Boltzmann-Grad scaling corresponding to $\gamma =\frac{n}{n-1}$, it is shown that the limiting distribution ${\phi }_{ϵ}$ of ${\tau }_{ϵ}$ is bounded from below by an expression of the form C/t, for some C> 0. A numerical study seems to indicate that asymptotically for large t, ${\phi }_{ϵ}\sim C/t$. This is an extension of a previous work [J. Bourgain et al., Comm. Math. Phys.190 (1998) 491-508]. As a consequence, it is proved that the linear Boltzmann type transport equation is inappropriate...

Consider the region obtained by removing from ${ℝ}^2$ the discs of radius $\epsilon$, centered at the points of integer coordinates $(a,b)$ with $b\not\equiv a(mod\ell )$. We are interested in the distribution of the free path length (exit time) ${\tau }_{\ell ,\epsilon }\left(\omega \right)$ of a point particle, moving from $(0,0)$ along a linear trajectory of direction $\omega$, as $\epsilon \to 0^{+}$. For every integer number $\ell \ge 2$, we prove the weak convergence of the probability measures associated with the random variables $\epsilon {\tau }_{\ell ,\epsilon }$, explicitly computing the limiting distribution. For $\ell =3$, respectively $\ell =2$, this result leads...

The Povzner equation is a version of the nonlinear Boltzmann equation, in which the collision operator is mollified in the space variable. The existence of stationary solutions in $L^1$ is established for a class of stationary boundary-value problems in bounded domains with smooth boundaries, without convexity assumptions. The results are obtained for a general type of collision kernels with angular cutoff. Boundary conditions of the diffuse reflection type, as well as the given incoming profile, are...