Seminario EDPs y Mecánica de Fluidos
Analysis of PDE Models in Natural Sciences: Micro-Electro-Mechanical Systems, Surface Plasmon Polaritons and Sintering
Ponente: Runan He (ICMAT)Fecha: jueves 21 de noviembre de 2024 - 11:30Lugar: Aula Naranja, ICMAT
Resumen:
The first part of the talk introduces the study of some mathematical models for a Micro-Electro-Mechanical System (MEMS) capacitor, consisting of a fixed plate and a flexible plate separated by a fluid. It investigates the wellposedness of solutions to the resulting quasilinear coupled systems, as well as the finite-time blow-up (quenching) of solutions. The models considered include a parabolic-dispersive system modelling the fluid flow under an elastic plate, a parabolic-hyperbolic system for a thin membrane, as well as an elliptic-dispersive system for quasistatic fluid flow under an elastic plate. Short-time existence, uniqueness and smoothness are obtained by combining wellposedness results for a single equation with an abstract semigroup approach for the system. Quenching is shown to occur, if the solution ceases to exist after a finite time.
The second part of the talk introduce 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 and proves 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. We also show that the real frequency exists in the nonlinear setting in the case of $PT$-symmetric materials.
The third part of the talk introduces the Mullins equation modelling sintering process. We show that the existence of the self-similar solution to the Mullins equation for large groove angle and the nonexistence of self-similar solution for small groove angle.
