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In this paper, we consider a non-isothermal electrokinetic model, which isderived from the Energetic Variational Approach. The charge transport isdescribed through the Poisson-Nernst-Planck equations with variabletemperature, and the heat flux satisfies the Fourier's law. ThisPoisson-Nernst-Planck-Fourier model satisfies both the first law and second lawof thermodynamics as well as the Onsager's reciprocal relations, thus it isthermodynamic-consistent. Finally, we prove the global well-posedness for thismodel under the smallness assumption of the initial data by the energy method.

Global Existence of the Non-isothermal Poisson-Nernst-Planck-Fourier System