When several reservoirs, initially in different equilibrium states, are brought into contact (directly or, possibly, mediated by a finite quantum system), it is to be expected that the state of the composite system will eventually converge to a non-equilibrium stationary state (NESS), characterized by constant energy and particle currents between reservoirs. The scattering approach, initiated by D. Ruelle, to modeling this physical setup in the operator algebra framework of quantum statistical mechanics is briefly described. In the body of the paper, a class of solvable microscopic quantum models is considered, in which the reservoirs are (Fermi or) Bose gases at thermal equilibrium and for which the dynamics, both of the isolated reservoirs and of the coupled system, are quasi-free automorphism groups of a suitable canonical (anti)commutation relation algebra. Within this class, the existence problem of the NESS is reduced to the scattering and spectral problems for the interacting and noninteracting "one-particle" automorphisms. Though this simplifying assumption restricts the contacts only to particle tunnelling between reservoirs, it allows to prove the existence, and derive analytically many properties, of stationary states in various physical instances. Recent results on the transport properties of quasi-free Fermions substantiating the Landauer-Büttiker formula for conductance and linear response theory (Green-Kubo formulas, Onsager relations, etc.) are reviewed. The case of Bose particles is more complex due to the phenomenon of Bose-Einstein condensation, implying non-uniqueness of the equilibrium state of the reservoirs. We derive for this case expressions of the currents in the NESS, including the ''Josephson super-current'' exhibiting a characteristic sin-dependence on the phase difference between condensates.

We survey the recent progress on the application of monotonicity methods (with respect to order) to the existence theory for various Boltzmann-like, nonlinear kinetic equations. To motivate the topic, we first provide several examples of Boltzmann models for complex systems, with similar monotonicity properties, which present interest in applications. These are Smoluchowski's coagulation equation, Povzner-like models with dissipative collisions and reactive collisions, respectively, a Boltzmann model for several chemical species (with reactions), and a von Neumann-Boltzmann quantum model. The common properties of the above models can be abstracted into a very general setting. One obtains a class of nonlinear evolution equations, formulated into an abstract Lebesgue space, for which one can state general criteria for the existence, uniqueness and positivity of global (in time) solutions. The proofs extend techniques that were initially developed in the more particular context of the space-homogeneous version of the classical Boltzmann equation. Finally we show how the abstract results can be applied to our examples of Boltzmann-like models.

We prove the analog of the Cwickel-Lieb-Rosenblum estimation for the number of negative eigenvalues of a relativistic Hamiltonian with magnetic field $B\inC^\infty_{\rm{pol}}(\mathbb R^d)$ and an electric potential $V\in L^1_{\rm{loc}}(\mathbb R^d)$, $V_-\in L^d(\mathbb R^d)\cap L^{d/2}(\mathbb R^d)$. Compared to the nonrelativistic case, this estimation involves both norms of $V_-$ in $L^{d/2}(\mathbb R^d)$ and in $L^{d}(\mathbb R^d)$. A direct consequence is a Lieb-Thirring inequality for the sum of powers of the absolute values of the negative eigenvalues.

The lecture presents the notion of approximate inertial manifold of a semi-dynamical system generated by a nonlinear evolution PDE (more precisely, a semilinear parabolic equation), as it appeared in the literature of the last twenty years. The localization of the attractors in the space of phases was a first interesting application field of the a.i.m.s. Besides, a.i.m.s found very interesting applications in the construction of some approximate solutions (and consequently in the numerical integration) of the nonlinear evolution problems. These are contained in the so-called nonlinear Galerkin and postprocessed Galerkin methods.

The paper deals with the numerical approximation of a class of nonlinear diffusion processes that includes the unsaturated water flow through porous media and the fast diffusion. The approximation method consists in the discretization of space derivative operators using the finite volume scheme and keeping the continuum time differentiation. Consequently, the solution of the partial differential equations is approximated by the solution of a system of ordinary differential equations. A scheme to approximate the diffusion and convective term such that one can obtain a quasi-monotone ODE system is defined. Further, it is proved that there exists a discrete comparison principle, the solutions of the discrete model are bounded and the upper and lower bounds are independent of the mesh size of triangulation. To perform the time numerical integration a class of implicit backward differentiation formulae with adaptive time step is used. Since the implicit schemes require a nonlinear solver a method that mixes Broyden method and an inexact Newton method is constructed. The performances of the new method are illustrated by some numerical results concerning the fast diffusion equation and water infiltration through a layered soil.

The paper deals with the extensions of approximation techniques of Nambu, Babovsky and Illner for the solutions of the classical Boltzmann equation to a nonlinear generalized Boltzmann-type system of equations solving nontrivial transport flows in dilute gas mixtures. First, one proves the global existence and uniqueness of solutions. Then a weak time-discretized version of equations for positive measures is provided. To obtain an algorithm, with small numerical effort (of order N\log N) stochastic methods are introduced. Finally a numerical approximation scheme, converging almost surely, in some sense, to the solutions of exact equations is provided.

The first part of the paper introduces diffusive models of water flow in saturated-unsaturated media, characterized by a space variation of the porosity. Then the analysis focuses on a model with mixed boundary conditions involving a flux on a part of the boundary and a nonhomogeneous Dirichlet condition corresponding to a singular situation (i.e., the blowing up diffusion coefficient) on the other part of the domain boundary. From the mathematical point of view, the problem resides in the study of a degenerate nonlinear variational inequality which can be reduced to a multivalued inclusion by an appropriate change of the unknown function. Finally, existence, uniqueness and other properties of the solution are established.