Global existence of the non-isothermal Poisson–Nernst–Planck–Fourier system

In this paper, we use the Energetic Variational Approach to derive a non-isothermal electrokinetic model. The charge transport is described through the Poisson–Nernst–Planck equations with variable temperature, and the heat flux satisfies the Fourier's law. This Poisson–Nernst–Planck–Fourier model satisfies both the first law and second law of thermodynamics as well as the Onsager's reciprocal relations, thus it is thermodynamic-consistent. Finally, we prove the global well-posedness for this model under the smallness assumption of the initial data by the energy method.