Generalized Huygens' principle for bipolar non-isentropic compressible Navier-Stokes-Poisson system in dimension three

Abstract

The generalized Huygens' principle for the non-isentropic bipolar Navier-Stokes-Poisson system in dimension three is given, which is different from the unipolar case. All of the previous estimates containing Huygens' waves on wave behaviors for compressible flow models rely heavily on the conservative structure, for instance, the Navier-Stokes system and even the isentropic bipolar Navier-Stokes-Poisson system. The reason is that the Huygens' waves decay slower than diffusion waves in Lp(R3)-space when 1<p<2. However, the non-isentropic case here is not conservative due to two energy equations, and moreover, it is not yet conservative even if one reformulates the original system as the isentropic case in [26]. The main contribution is that we first solve the nonlinear problem when there exists the Huygens' wave in the Green's function but the nonlinear system is not conservative, which is achieved by establishing refined pointwise estimates for the Green's function and nonlinear convolution estimates.