Semi-discretización de problemas parabólicos no lineales: resultados de convergencia y comportamiento asintótico

The main goal of the present project is to study semi-discrete numerical schemes for linear and nonlinear parabolic equations preserving important features such as conservation of mass, tim monotonicity of thermodynamical functionals (for example the free energy or the entropy), and positivity of the solutions. The studied equations include porous medium equations, semiconductor models, and cross-diffusion population models. In recent years the entropy dissipation technique has been proved to be a very powerful tool in order to prove the well-posedness and long-time behaviour of continuous linear and nonlinear diffusion equations (see, e.g. [2, 6, 16]). Concerning second-order equations, usually Fokker-Planck-type or porous medium equations have been studied as model equations. Also fourth-order parabolic equations have been studied, modelling quantum systems [15] or thin film flow [4]. Exponential decay rates for the solutions of these models have been obtained. However, there are much less results for numerical schemes of these equations. For the numerical computations it is of great importance to preserve the special features of the continuous solutions, like positivity, conservation of mass, time monotonicity of so-called entropy functionals. First results have been obtained by, e.g. [3, 7, 13]. The main objective of this project is to develop temporal and spatial semi-discretizations of nonlinear second-order and fourth-order equations, which preserve the special continuous features. In the following we describe the models in more detail. Fourth-order parabolic equations attracted recently a lot of attention in the mathematical literature. These equations are used for the modelling of surface dominated motion of thin viscous films and spreading droplets or plasticity, in certain spin systems and quantum semiconductor devices. The main feature of these models is that it allows for positive or non-negative solutions(...)