On absorbing set for 3D Maxwell--Schr\"odinger damped driven equations in bounded region

TL;DR

This paper studies the 3D Maxwell--Schrödinger equations with damping and driving forces in a bounded domain, establishing new a priori estimates that guarantee the existence of global solutions and bounded absorbing sets.

Contribution

It introduces novel Sobolev-type estimates for the magnetic Schrödinger operator, leading to the proof of global existence and bounded absorbing sets for the equations.

Findings

Existence of global finite energy weak solutions.

Establishment of bounded absorbing sets.

Development of new Sobolev estimates for magnetic Schrödinger operator.

Abstract

We consider the 3D damped driven Maxwell--Schr\"odinger equations in a bounded region under suitable boundary conditions. We establish new a priori estimates, which provide the existence of global finite energy weak solutions and bounded absorbing set. The proofs rely on the Sobolev type estimates for magnetic Schr\"odinger operator.