Space-time estimates of the 3D bipolar compressible Navier-Stokes-Poisson system with unequal viscosities.

The space-time behavior for the Cauchy problem of the 3D compressible bipolar Navier-Stokes-Poisson (BNSP) system with unequal viscosities is given. The space-time estimate of the electric field ∇ϕ = ∇(−Δ)⁻¹(n − Z_ρ) is the most important in deducing generalized the Huygens' principle for the BNSP system and it requires proving that the space-time estimate of n − Z_ρ only contains the diffusion wave due to the singularity of the operator ∇(−Δ)⁻¹. A suitable linear combination of unknowns reformulating the original system into two small subsystems for the special case (with equal viscosities) in Wu and Wang (2017) is so crucial for both linear analysis and nonlinear estimates, especially for the space-time estimate of ∇ϕ. However, the benefits from this reformulation will not exist any longer for general cases. Here, we study an 8×8 Green's matrix directly. More importantly, each entry in Green's matrix contains wave operators in the low frequency part, which will generally produce the Huygens' wave; as a result, one cannot achieve that the space-time estimate of n − Z_ρ only contains the diffusion wave as before. We overcome this difficulty by taking more detailed spectral analysis and developing new estimates arising from subtle cancellations in Green's function. [ABSTRACT FROM AUTHOR]