Pointwise space-time estimates of 3D bipolar compressible Navier-Stokes-Poisson system with unequal viscosities

Abstract

Space-time behaviors for 3D compressible bipolar Navier-Stokes-Poisson system (BNSP) with unequal viscosities are given. The space-time estimate of electric field $\nabla\phi$ is the most important thing when deducing generalized Huygens' principle for BNSP since this estimate only can be obtained by $\nabla\phi=\frac{\nabla}{\Delta}(\rho-n)$ from the Poisson equation. Thus, it requires to prove that the space-time estimate of $\rho-n$ only contains diffusion wave. The appearance of these unequal coefficients results that one cannot follow ideas for the special case, where the original system was rewritten as a compressible NS system and a compressible (unipolar) NSP system after a linear combination of unknowns. This linear combination brings special structure for nonlinear terms, and this structure was also used to get desired space-time estimate for $\rho-n$. Moreover, Green's function of the subsystem NSP does not contain Huygens wave is equally important in [36]. However, for the general case, the benefits from this linear combination will not exist any longer. First, we have to directly consider an $8\times8$ Green's matrix of the original system. Second, all of entries in Green's function in low frequency actually contain wave operators. This generally produces the Huygens' wave for each entry in Green's function, as a result, one cannot achieve that the space-time estimate of $\rho-n$ only contains the diffusion wave as usual. We overcome this difficulty by taking more detailed spectral analysis and developing new estimates arising from subtle cancellations in Green's function. Third, due to loss of the special structure of nonlinear terms from the linear combination, we shall develop new nonlinear convolution estimates such that we can ultimately obtain the expected space-time estimate for $\nabla\phi$ and further verify the generalized Huygens' principle.