Asymptotic behavior of solutions to the compressible bipolar Euler–Maxwell system in $\mathbb{R}^3$

We study the large time behavior of solutions near a constant equilibrium state to the compressible bipolar Euler–Maxwell system in $\mathbb{R}^3$. We first refine the global existence of solutions by assuming that the initial data is small in the $H^3$ norm, but its higher order derivatives could be large. If, further, the initial data belongs to $\dot{H}^{-s} (0 \leqslant s \lt 3/2)$ or $\dot{B}^{-s}_{2,\infty} (0 \lt s \leqslant 3/2)$, then we obtain the various time decay rates of the solution and its higher order derivatives. As an immediate byproduct, the $L^p - L^2 (1 \leqslant p \leqslant 2)$ type of the decay rates follows without requiring the smallness for $L^p$ norm of initial data. So far, our decay results are most comprehensive ones for the bipolar Euler–Maxwell system in $\mathbb{R}^3$.

Keywords

compressible bipolar Euler–Maxwell system, time decay rates, ensergy method, interpolation

2010 Mathematics Subject Classification

35A01, 35B40, 35Q35, 35Q61, 82D10

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Published 19 August 2015