Quasi-Linear Second-Order Parabolic Equations With Many Independent Variables

Abstract

We present a model for microwave photoconductivity in two-dimensional electron systems (2DESs) in a magnetic field at microwave frequencies lower than the electron cyclotron frequency when the intra-Landau level (LL) transitions dominate. Using this model, we explain the effect of the decrease in the 2DES dissipative conductivity (and resistivity) and smearing of its Shubnikov-de Haas oscillations caused by microwave radiation observed recently. The model invokes the concept of suppression of elastic impurity scattering of electrons by the microwave electric field. We calculated the dependence of the 2DES conductivity associated with intra-LL transitions as a function of the radiation and cyclotron frequencies and microwave power. We take into account the effect of plasma reflection of microwaves from 2DES resulting in a distinction between the ac microwave electric field acting on electrons and that generated by a microwave source. The obtained dependences are consistent with the results of recent experimental observations.

Corrected Operator Splitting for Nonlinear Parabolic Equations

“…Consequently, in order to show that the limit of a converging sequence is the unique classical solution to (1), it is sufficient to demonstrate that the limit satisfies (8). We refer to [28,29] for a survey of the mathematical theory of nonlinear parabolic equations such as (1) and (2). In what follows, · p denotes the usual L p norm (p = 1, .…”

“…Consequently, we choose to work with (1) in this paper. Existence and uniqueness of a classical solution to (1) is well known; see, e.g., [28,29]. Furthermore, the notion of a classical solution coincides with the notion of a weak solution for parabolic equations such as (1); see [28].…”

Analyse mathématique d'un systéme de transport-diffusion-réaction modélisant la restauration biologique d'un milieu poreux

“…Dérnonstration du théor~me 2 Suivant, l'idée utilisée dans [4] et [13], notre méthode consiste it découpler et it linéariser les équations (8)... (11), en introduisant un retard en temps dans les termes non-linéaires; on réallse ainsi un algorithme qui calcule la pression et la vitesse par l'équation de Darcy en prenant la concentration 5 sohution de l'équation d'advection diffusíon réaction it l'intervalle de ternps pr&édent. Qn utilise cette vitesse pour évaluer lea concentrations solutions des équations linéarisées sur l'intervalle de temps suívañt.…”

“…Ensuite, pour chaque u en résaud les équations par la métImode de Galerkin. Enfin, gráce aux estimations a priori du mame type que (13)... (24), on peut passer it la limite et condure (voir aussi [13]). …”

“…Cette technique présente de nombreux avantages car elle est écologique, súre, rapide et économique 13]. La restauration bialogique est un phénoméne complexe íncluant le transpoñ et l'adsorption des substrats en solution et l'interaction entre les bactéries et les polluants.…”