Existence analysis of a three-species memristor drift-diffusion system coupled to electric networks

TL;DR

This paper proves the existence of global weak solutions for a complex three-species memristor model coupled with electric networks, using advanced mathematical techniques to ensure solutions are bounded and positive.

Contribution

It introduces a rigorous mathematical existence proof for a coupled memristor-electrical network system involving drift-diffusion and algebraic equations.

Findings

Existence of global weak solutions established.

Solutions are shown to be bounded and strictly positive.

The proof utilizes fixed-point theorem and energy inequalities.

Abstract

The existence of global weak solutions to a partial-differential-algebraic system is proved. The system consists of the drift-diffusion equations for the electron, hole, and oxide vacancy densities in a memristor device, the Poisson equation for the electric potential, and the differential-algebraic equations for an electric network. The memristor device is modeled by a two-dimensional bounded domain, and mixed Dirichlet-Neumann boundary conditions for the electron and hole densities as well as the potential are imposed. The coupling is realized via the total current through the memristor terminal and the network node potentials at the terminals. The network equations are decomposed in a differential and an algebraic part. The existence proof is based on the Leray-Schauder fixed-point theorem, a priori estimates coming from the free energy inequality, and a logarithmic-type Gagliardo-Nirenberg inequality. It is shown, under suitable assumptions, that the solutions are bounded and strictly positive.