## Decay estimates of solutions to the compressible Euler-Maxwell system in $$\mathbb{R}^3$$.(English)Zbl 1296.83018

Summary: We study the large time behavior of solutions near a constant equilibrium to the compressible Euler-Maxwell system in $$\mathbb{R}^3$$. We first refine a global existence theorem by assuming that the $$H^3$$ norm of the initial data is small, but the higher order derivatives can be arbitrarily large. If the initial data belongs to $$\dot{H}^{- s}$$ ($$0 \leq s < 3 / 2$$) or $$\dot{B}_{2, \infty}^{- s}$$ ($$0 < s \leq 3 / 2$$), by a regularity interpolation trick, we obtain the various decay rates of the solution and its higher order derivatives. As an immediate byproduct, the usual $$L^p$$-$$L^2(1 \leq p \leq 2)$$ type of the decay rates follows without requiring that the $$L^p$$ norm of initial data is small.

### MSC:

 83C22 Einstein-Maxwell equations 82D37 Statistical mechanics of semiconductors 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 35Q35 PDEs in connection with fluid mechanics 35B40 Asymptotic behavior of solutions to PDEs 83C15 Exact solutions to problems in general relativity and gravitational theory 83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems) 85A30 Hydrodynamic and hydromagnetic problems in astronomy and astrophysics
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