arXiv subjects

Runan He

Publications and source records attributed to Runan He.

8 recordsLinked to original sources

arXiv2026

Analysis and Numerics of a Stationary Drift-Diffusion Model for Electrical Discharge in MEMS

This work presents the analysis and numerical simulation of a stationary drift-diffusion model for electrical discharge in micro-electro-mechanical systems (MEMS). The model couples Poisson's equation for the electrostatic potential with continuity equations for positive ions and electrons, incorporating a Townsend-type impact ionization source term that depends exponentially on the electric field magnitude. We prove the existence of weak solutions under physically relevant assumptions and establish uniform bounds on the carrier densities. The proof relies on a regularization-approximation scheme with truncated nonlinearities, monotone operator theory (Browder-Minty), iterative energy estimates, and Stampacchia-type truncation arguments. We further develop a robust finite element solver to simulate the carrier density and electrostatic potential profiles for several geometries, including two-dimensional domains and a three-dimensional axisymmetric geometry.

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arXiv2026

Analysis of a Model for Electrical Discharge in MEMS

We study the local well-posedness of the solution to a coupled nonlinear elliptic-parabolic system which models electrical discharge in a Micro-Electro-Mechanical System (MEMS). A simple MEMS capacitor device contains two plates acting as the capacitor's electrodes, one of which is flexible, and which are separated by a narrow gas-filled gap. In the event of the flexible plate approaching the other, electrical discharge can occur. This is modelled here by two parabolic equations, for densities of electrons and positive ions, and an elliptic equation for electric potential. We show the local-in-time existence of a weak solution for the coupled system. Compactness techniques, used previously in the study of drift-diffusion equations, are employed in our proof.

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arXiv2025

Polychromatic Localized Waves with Complex Frequencies in Nonlinear Maxwell Equations with Material Dispersion

We study the existence of polychromatic solutions of cubically nonlinear Maxwell equations in the whole space and with dispersive media, i.e., with a time delayed polarization. Due to the complex nature of the dielectric function, the frequencies are complex, resulting in a decay in time. The geometry is that of a waveguide in $x$ with the propagation direction being $y$ and the solutions are localized in $x$ and have a transverse magnetic polarization. These are often referred to as breathers. They are given as a Fourier series in $y$ and $t$ with the leading frequency $\omega$ being an eigenvalue of a corresponding operator pencil on $\mathbb{R}$ (in the $x$ variable). Each term in the series corresponds to a different temporal decay rate or a different frequency. The series is constructed iteratively via a sequence of linear ordinary differential equations. Our general result provides the existence under some assumptions on the spectrum and on estimates of the resolvent of the corresponding linear operator. We also produce an example of a waveguide given by the interface of two spatially homogeneous physically relevant media for which these assumptions are satisfied. For such an interface setting the constructed solutions correspond to nonlinear polychromatic surface plasmons.

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arXiv2024

Existence and uniqueness for a coupled parabolic-hyperbolic model of MEMS

Local well-posedness for a nonlinear parabolic-hyperbolic coupled system modelling Micro-Electro-Mechanical System (MEMS) is studied. The particular device considered is a simple capacitor with two closely separated plates, one of which has motion modelled by a semi-linear hyperbolic equation. The gap between the plates contains a gas and the gas pressure is taken to obey a quasi-linear parabolic Reynolds' equation. Local-in-time existence of strict solutions of the system is shown, using well-known local-in-time existence results for the hyperbolic equation, then H\"{o}lder continuous dependence of its solution on that of the parabolic equation, and finally getting local-in-time existence for a combined abstract parabolic problem. Semigroup approaches are vital for the local-in-time existence results.

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arXiv2023

Bifurcation and Asymptotics of Cubically Nonlinear Transverse Magnetic Surface Plasmon Polaritons

Linear Maxwell equations for transverse magnetic (TM) polarized fields support single frequency surface plasmon polaritons (SPPs) localized at the interface of a metal and a dielectric. Metals are typically dispersive, i.e. the dielectric function depends on the frequency. We prove the bifurcation of localized SPPs in dispersive media in the presence of a cubic nonlinearity and provide an asymptotic expansion of the solution and the frequency. The problem is reduced to a system of nonlinear differential equations in one spatial dimension by assuming a plane wave dependence in the direction tangential to the (flat) interfaces. The number of interfaces is arbitrary and the nonlinear system is solved in a subspace of functions with the $H^1$-Sobolev regularity in each material layer. The corresponding linear system is an operator pencil in the frequency parameter due to the material dispersion. Because of the TM-polarization the problem cannot be reduced to a scalar equation. The studied bifurcation occurs at a simple isolated eigenvalue of the pencil. For geometries consisting of two or three homogeneous layers we provide explicit conditions on the existence of eigenvalues and on their simpleness and isolatedness. Real frequencies are shown to exist in the nonlinear setting in the case of PT-symmetric materials. We also apply a finite difference numerical method to the nonlinear system and compute bifurcating curves.

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arXiv2023

Wellposedness of a nonlinear parabolic-dispersive coupled system modelling MEMS

In this paper we study the local wellposedness of the solution to a non-linear parabolic-dispersive coupled system which models a Micro-Electro-Mechanical System (MEMS). A simple electrostatically actuated MEMS capacitor device has two parallel plates separated by a gas-filled thin gap. The nonlinear parabolic-dispersive coupled system modelling the device consists of a quasilinear parabolic equation for the gas pressure and a semilinear plate equation for gap width. We show the local-in-time existence of strict solutions for the system, by combining a local-in-time existence result for the dispersive equation, H\"{o}lder continuous dependence of its solution on that of the parabolic equation, and then local-in-time existence for a resulting abstract parabolic problem. Semigroup approaches are vital for both main parts of the problem.

math.AP↗

arXiv2023

Wellposedness of an elliptic-dispersive coupled system for MEMS

In this work, we study the local wellposedness of the solution to a nonlinear elliptic-dispersive coupled system which serves as a model for a Micro-Electro-Mechanical System (MEMS). A simple electrostatically actuated MEMS capacitor device consists of two parallel plates separated by a gas-filled thin gap. The nonlinear elliptic-dispersive coupled system modelling the device combines a linear elliptic equation for the gas pressure with a semilinear dispersive equation for the gap width. We show the local-in-time existence of strict solutions for the system, by combining elliptic regularity results for the elliptic equation, Lipschitz continuous dependence of its solution on that of the dispersive equation, and then local-in-time existence for a resulting abstract dispersive problem. Semigroup approaches are key to solve the abstract dispersive problem.

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arXiv2022

Quenching for a semi-linear wave equation for MEMS

We consider the formation of finite-time quenching singularities for solutions of semi-linear wave equations with negative power nonlinearities, as can model micro-electro-mechanical systems (MEMS). For radial initial data we obtain, formally, the existence of a sequence of quenching self-similar solutions. Also from formal asymptotic analysis, a solution to the PDE which is radially symmetric and increases strictly monotonically with distance from the origin quenches at the origin like an explicit spatially independent solution. The latter analysis and numerical experiments suggest a detailed conjecture for the singular behaviour.

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