Global existence and time decay rates for the 3D bipolar compressible Navier-Stokes-Poisson system with unequal viscosities

Abstract

The space-time behaviors for Cauchy problem of 3D compressible bipolar Navier-Stokes-Poisson system (BNSP) with unequal viscosities and unequal pressure functions are given. As we know, the space-time estimate of electric field ∇φ is the most important one in deducing generalized Huygens' principle for BNSP since this estimate only can be obtained by the relation ∇φ = ∇ ∆ (Zρ−n) from the Poisson equation. Thus, it requires to prove that the space-time estimate of Zρ−n only contains diffusion wave. The appearance of these unequal coefficients results that one cannot follow ideas for the special case in Wu-Wang [36], where the original system was rewritten as a compressible Navier-Stokes system (NS) and a unipolar compressible Navier-Stokes-Poisson system (NSP) after a suitable linear combination of unknowns. Additionally this linear combination brings some special structure for nonlinear terms, and this special structure was also used to get desired space-time estimate for Zρ − n. Moreover, Green's function of the subsystem NSP does not contain Huygens' wave is equally important in the proof in [36]. However, for the general case here, the benefits from this linear combination will not exist any longer. First, we have to directly consider an 8 × 8 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 Zρ−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 ∇φ and further verify the generalized Huygens' principle for the general case.

Optimal large time behavior of the compressible Bipolar Navier--Stokes--Poisson system with unequal viscosities

“…This section is devoted to prove the optimal decay rates of the solution stated in Theorem 1.1. First by the general energy estimate method, we can derive the following energy inequality, whose proof can also be seen in [34]: Proposition 3.1. Under the assumption (1.8) of Theorem 1.1, the Cauchy problem (2.3) admits a unique globally classical solution (̺ 1 , u 1 , ̺ 2 , u 2 ) such that for any t ∈ [0, ∞),…”

“…Thus in order to obtain a priori estimates of the solutions, one can apply the similar arguments as in [3,22,23] for the Navier-Stokes system and in [20,30] for the UNSP system. Recently, under the assumption that the initial perturbation is small in H l (R 3 ) with l ≥ 3, Wu-Zhang-Zhang [34] established the global existence for the BNSP system with unequal viscosities, i.e.,…”

“…Remark 1. The global existence of the Cauchy problem (1.1)-(1.2) with unequal viscosities (1.3) under the small initial perturbation assumption has been proved in [34] by the standard continuity argument. In this paper, we focus on the upper-lower time decay rates of the solution as well as allorders spatial derivatives, and the explicit influences of the electric field on the qualitative behaviors of solutions.…”

Optimal temporal decay rate of the non‐isentropic compressible Navier–Stokes–Poisson system

“…There also exist rich results concerning the asymptotic behavior of the solution to the NSP system (refer to [22][23][24][25][26][27][28][29][30][31][32] and references therein). More precisely, Wang and Wu [29] claimed the point-wise profile of the solution contains only the D-wave due to the effect of the nonlocal term.…”

Optimal large time behavior of the compressible bipolar Navier–Stokes–Poisson system

“…Thus in order to obtain a priori estimates of the solutions, one can employ similar arguments as in [3,22,23] for the Navier-Stokes system and in [20,29] for the UNSP system. Recently, under the assumption that the initial perturbation is small in H l (R 3 ) with l ≥ 3, Wu-Zhang-Zhang [33] established the global existence for the BNSP system with unequal viscosities, i.e.,…”