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The Cauchy problem of the bipolar Navier–Stokes–Poisson system (1.1) in dimension three is considered. We obtain the pointwise estimates of the time-asymptotic shape of the solution, which exhibit a generalized Huygens’ principle as the Navier–Stokes system. This phenomenon is the most important difference from the unipolar Navier–Stokes–Poisson system. Due to the non-conservative structure of the system (1.1) and the interplay of two carriers which counteract the influence of the electric field (a nonlocal term), some new observations are essential for the proof. We fully use the conservative structure of the system for the total density and total momentum, and the mechanism of the linearized unipolar Navier–Stokes–Poisson system together with the special form of the nonlinear terms in the system for the difference of densities and the difference of momentums. Lastly, as a byproduct, we extend the usual $${L^2({\mathbb{R}}^3)}$$ L 2 ( R 3 ) -decay rate to the $${L^p({\mathbb{R}}^3)}$$ L p ( R 3 ) -decay rate with $${p > 1}$$ p > 1 and also improve former decay rates.
Archive for Rational Mechanics and Analysis – Springer Journals
Published: Jun 10, 2017
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