Existence of weak solutions of the drift diffusion model coupled with Maxwell's equations
The paper is concerned with the system \[ {\partial\rho_k\over\partial t}= -\text{div }J_k-R,\quad k=1,2,\tag{1} \] \[ {\partial E\over\partial t}= \text{curl }H+ J_2- J_1,\quad {\partial H\over\partial t}= -\text{curl }E,\tag{2} \] \[ \text{div }E= \rho_1- \rho_2+ C,\quad \text{div }H= 0,\tag{3} \] where \(\rho_1\), \(\rho_2\) = densities of holes and electrons, respectively, \(J_1=-D_1\nabla \rho_1+ \mu_1\rho_1 E\), \(J_2= -D_2\nabla\rho_2- \mu\rho_2 E\) are the current densities, \(R\) = recombination/generation of carriers, and \(E\), \(H\) = electromagnetic field. System (1)--(3) is considered in \(\Omega\times (0,+\infty)\) (\(\Omega\subset\mathbb{R}^3\) a bounded Lipschitz domain) and completed by mixed boundary conditions on \(\rho_1\), \(\rho_2\), \(E\), and \(H\) and initial conditions. The author proves the existence of a weak solution to (1)--(3) on any interval \((0,T)\) \((0<T<+\infty)\) by solving a regularized system via the contraction mapping principle, establishing a-priori-estimates on the solutions of the regularized system and then passing to the limit by using compactness arguments.
scientific article; zbMATH DE number 1005592