In this talk, I will present the Nernst-Planck-Boussinesq (NPB) system, a coupled model that combines the Nernst-Planck equations with the Boussinesq approximation to describe ionic electrodiffusion in an incompressible fluid under non-isothermal conditions. A key feature of the NPB system is the nonlinear structure of the electromigration term, which is influenced by the reciprocal of the temperature. This distinguishes the NPB system from other electrodiffusion models, such as the Nernst-Planck-Navier-Stokes system, and introduces unique mathematical challenges. I will discuss several analytical properties of the system, focusing on the global existence of weak solutions in three dimensions, as well as the long-time behavior of these solutions, including their exponential convergence to steady states.
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