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Global classical solutions to the compressible Euler-Maxwell equations

In this paper, we consider the compressible Euler-Maxwell equations arising in semiconductor physics, which take the form of Euler equations for the conservation laws of mass density and current density for electrons, coupled to Maxwell's equations for self-consistent electromagnetic field. We study the global well-posedness in critical spaces and the limit to zero of some physical parameters in the scaled Euler-Maxwell equations. More precisely, using high- and low-frequency decomposition methods, we first construct uniform (global) classical solutions (around constant equilibrium) to the Cauchy problem of Euler-Maxwell equations in Chemin-Lerner's spaces with critical regularity. Furthermore, based on Aubin-Lions compactness lemma, it is justified that the (scaled) classical solutions converge globally in time to the solutions of compressible Euler-Poisson equations in the process of non-relativistic limit and to that of drift-diffusion equations under the relaxation limit or the combined non-relativistic and relaxation limits.