Global Existence for a Drift-Diffusion Model of Electrical Discharge in MEMS

Abstract

We study a coupled elliptic-parabolic system modelling electrical discharge in a micro-electro-mechanical (MEMS) capacitor: a Poisson equation for the electrostatic potential, coupled to drift-diffusion equations for the ion and electron densities with a Townsend ionization source whose rate per electron grows linearly in the electric field. Gimperlein, He and Lacey proved local well-posedness and left open whether the densities blow up in finite time. We first prove a continuation criterion in a weak norm: a finite-time singularity forces the space charge to be unbounded in H^1_D, the dual of the space of H^1 functions vanishing on the electrodes, so blow-up cannot be quasi-neutral. We then exclude it altogether when the two diffusivities coincide and the vacuum field is bounded: the weak solution is then global in time. To our knowledge this is the first global existence result without smallness assumptions for a drift-diffusion-Poisson model of gas discharge whose ionization rate is unbounded in the field. The proof uses the electrostatic energy and the number of carriers in place of the usual free energy: the ionization source cancels in the energy balance, the Joule heating is dissipative, and the two quantities close through a linear Gronwall inequality. Numerical experiments on a discharge in a micrometre gap, with the Townsend constants and the mobilities of air, provide evidence for the extension to unequal diffusivities and bear on the existence of a breakdown threshold.