On a nonlocal model of non-isothermal phase separation

A nonlocal model of non-isothermal phase separation in binary alloys is presented. The model is deduced from a free energy with a nonconvex part taking into account nonlocal particle interaction. The model consists of a system of second order parabolic evolution equations for heat and mass, coupled by nonlinear drift terms and a state equation which involves a nonlocal interaction potential. The negative entropy turns out to be Lyapunov functional of the system and yields the key estimate for proving global existence and uniqueness results and for analyzing the asymptotic behaviour as time goes to infinity

A dissipative discretization scheme for a nonlocal phase segregation model

We are interested in finite volume discretization schemes and numerical solutions for a nonlocal phase segregation model, suitable for large times and interacting forces. Our main result is a scheme with definite discrete dissipation rate proportional to the square of the driving force for the evolution, i. e., the discrete antigradient of the chemical potential v. Steady states are characterized by constant v and satisfy a nonlocal stationary equation. A numerical bifurcation analysis of that stationary equation explains the observed global behavior of numerically computed trajectories of the evolution equation. For strong interaction forces the model shows steady states distinguished by small deformations of the 'mushy region' or 'interface states'. One essential open question in the discrete case is the global boundedness of v

Global behaviour of reaction-diffusion system modelling chemotaxis

Using Lyapunov functionals the global behaviour of the solutions of a reaction-diffusion system modelling chemotaxis is studied for bounded piecewise smooth domains in the plane. Geometric criteria can be given that this dynamical system tends to a (not necassarity trivial) stationary state

On unique solvability of nonlocal drift-diffusion type problems

We prove global existence and uniqueness of bounded weak solutions to Cauchy--Neumann problems for degenerate parabolic equations with drift terms determined by integral equations instead of by elliptic boundary problems as in the corresponding local case. Such problems arise as mathematical models of various transport processes driven by gradients of local particle concentrations and nonlocal interaction potentials. Examples are transport of charge carriers in semiconductors and phase separation processes in alloys

Existence and uniqueness results for reaction-diffusion processes of electrically charged species

We study initial-- boundary value problems for elliptic--parabolic systems of nonlinear partial differential equations describing drift--diffusion processes of electrically charged species in N--dimensional bounded Lipschitzian domains. We include Fermi--Dirac statistics and admit nonsmooth material coefficients. We prove existence and uniqueness of bounded global solutions

A descent method for the free energy of multicomponent systems

Equilibrium distributions of a multicomponent system minimize the free energy functional under the constraint of mass conservation of the components. However, since the free energy is not convex in general, one tries usually to characterize and to construct equilibrium distributions as steady states of an adequate evolution equation (for example, the nonlocal Cahn-Hilliard equation for binary alloys). In this work a direct descent method for nonconvex functionals is established and applied to phase separation problems in multicomponent systems and image segmentation

To the uniqueness problem for nonlinear elliptic equations

We prove existence, boundedness and uniqueness of solutions to nonlinear elliptic boundary of second order under nonstandard assumptions with respect to the coefficients. Our assumptions are natural in view of drift diffusion processes for example in semiconductors and chemotaxi

Thermodynamics-based modeling edge-emitting quantum well lasers

This paper describes the modeling and the simulation of edge-emitting quantum well lasers, based on the drift-diffusion equations and equations for the optical field. By applying fundamental thermodynamic principles as the maximum entropy principle and the principle of local thermal equilibrium we derive a self-consistent energy transport model which can be proven to meet the thermodynamic requirements. It's numerical solution is discussed explicitly, by starting from the discretization procedure and by ending up with the iteration scheme. As an example, we demonstrate the simulation of a long-wavelength ridge-waveguide multi-quantum well laser

Thermodynamic design of energy models of semiconductor devices

In this preprint a system of evolution equations for energy models of a semiconductor device is derived on an deductive way from a generally accepted expression for the free energy. Only first principles like the entropy maximum principle and the principle of partial local equilibrium are applied. Particular attention is paid to include the electrostatic potential self-consistently. Dynamically ionized trap levels and models with carrier temperatures are regarded. The system of evolution equations is compatible with the corresponding entropy balance equation that contains a positively definite entropy production rate