Mathematical Models and Methods in Applied Sciences

In this paper we deal with a Cauchy–Dirichlet quasilinear parabolic problem containing a gradient lower order term; namely, u_t - Δu + |u|^{2 γ-2}u |∇u|² = |u|^p-2u. We prove that if p ≥ 1, γ ≥ ½ and p < 2 γ + 2, then there exists a global weak solution for all initial data in L¹ (Ω). We also see that there exists a non-negative solution if the initial datum is non-negative.

In this paper, we present a multi-fluid ionization model. We prove that this stationary, mono-dimensional model has a maximal solution which is not global at variance with the mono-species case and we present a numerical method for solving this highly singular system of ordinary differential equations. Numerical results are compared with those obtained for other models.

Starting from a mesoscopic principle of moment conservation, discrete Boltzmann collision operators J^h are constructed, which both converge to bounded collision operators J_Ω and have the same collision invariants as the original Boltzmann collision operator J. The crucial point of this construction is the application of a weak formulation of the gain operator to remove the post-collision velocities from it as well as the development of moment conserving integration formulas for the approximation of surface integrals over the unit sphere. Finally two applications for the discrete operators are presented.

The general integral equations used to formulate all the balance laws of thermodynamics of charged continuous systems are analyzed. In particular, simpler (with respect to those known in the literature) procedures are proposed for the localization process leading from the integral balance laws to differential equations and jump conditions.

The design problem for semiconductor devices is studied via an optimal control approach for the standard drift–diffusion model. The solvability of the minimization problem is proved. The first-order optimality system is derived and the existence of Lagrange-multipliers is established. Further, estimates on the sensitivities are given. Numerical results concerning a symmetric n–p-diode are presented.

In the external radiation therapy the source of radiation is from outside. The healthy tissue and some organs, called critical organs which are quite intolerable for radiation, are always irradiated, too. Therefore, the careful treatment plan has to be constructed to ensure high and homogeneous dose in the tumor, but on the other hand to spare the normal tissue and critical organs possibly well.

In the radiation therapy treatment planning one tries to optimize the dose distribution in the way that the above aim is satisfied. The dose distributions can be generated with different techniques. The most recent of them is the so-called multileaf collimator (MLC) delivery technique. Calculation of the dose distribution demands some dose calculation model. The paper gives a model and theoretical basis of planning applying the Boltzmann-transport equation in dose calculation and MLC delivery technique. The existence of solutions and the optimal treatment planning are considered. A preliminary artificial computer simulation is included.

We consider the compressible barotropic Navier–Stokes system in one dimension, with a nonmonotone equation of state and self-gravitation. We solve an associated "fixed-free" boundary problem and prove asymptotic properties of the unique globally defined solution for large time. We also comment on a related model of quantum fluid describing the dynamics of nuclear matter with zero temperature and Coulomb effects.