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\(L^p - L^q - L^r\) estimates and minimal decay regularity for compressible Euler-Maxwell equations. (English. French summary) Zbl 1330.35052

The authors consider isentropic Euler-Maxwell equations in \([0,\infty) \times \mathbb{R}^3\), where the energy equation is replaced with a state equation of the pressure density relation. In several space dimensions, the question of global (in time) weak solutions is still open. The global (in time) existence and large time behavior of smooth solutions have been studied. In the present paper, the authors are interested in deriving time decay of classical solutions. The first step is to rewrite the given system as a linearized perturbation form around equilibrium states. Using phase space analysis, optimal decay estimates are proved. The authors observe a regularity-loss of solutions comparing with data because some additional regularity is used to derive optimal decay estimates. This observation is transferred to the given nonlinear system. The phase space analysis consists of special \(L^p-L^q-L^r\) estimates for Fourier multipliers localized to small and large frequencies as well. Then, the authors use the energy method in the Fourier space. They prove a pointwise estimate of the Fourier transform of classical solutions to the Cauchy problem for the linearized system. Finally, a time-weighted energy functional is introduced and a nonlinear inequality for this functional is proved. All these preparations lead immediately to a stability result and an optimal decay estimate for classical solutions around an equilibrium state.

MSC:

35B45 A priori estimates in context of PDEs
35B35 Stability in context of PDEs
35L40 First-order hyperbolic systems
35B40 Asymptotic behavior of solutions to PDEs
82D10 Statistical mechanics of plasmas
35Q31 Euler equations
35Q61 Maxwell equations
35A09 Classical solutions to PDEs
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