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Global existence for the Euler-Maxwell system. (Existence globale pour le système d’Euler-Maxwell.) (English. French summary) Zbl 1311.35195

In this paper, the authors prove global existence of small solutions to the Euler-Maxwell system set in the whole three-dimensional space. This system reads \[ \begin{cases} \rho \left( \partial_t u + u\cdot \nabla u \right) = -\frac{p'(\rho)}{m} \nabla \rho - \frac{e \rho}{m}\Big( E + \frac{1}{c} u \times B \Big) \\ \partial_t \rho + \nabla \cdot (\rho u) = 0 \\ \partial_t B + c \nabla \times E = 0 \\ \partial_t E - c \nabla \times B = 4 \pi e \rho u \\ \nabla \cdot E = 4 \pi e (\bar \rho - \rho) \\ \nabla \cdot B = 0 \\ (u,\rho,E,B)(t=0) = (u_0,\rho_0,E_0,B_0), \end{cases} \] where the unknown functions are \((u,\rho,E,B)\) and \((c,e,m, \bar \rho)\) are constants.
The global existence with scattering is obtained by combining the space-time resonance method (to obtain decay) and energy estimates (to control high frequencies).

MSC:

35Q31 Euler equations
35L03 Initial value problems for first-order hyperbolic equations
35L60 First-order nonlinear hyperbolic equations
35Q60 PDEs in connection with optics and electromagnetic theory
37L50 Noncompact semigroups, dispersive equations, perturbations of infinite-dimensional dissipative dynamical systems
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