论文标题
3D双极压缩的Navier-Stokes-poisson系统的点上时空估计值不平等的粘度
Pointwise space-time estimates of 3D bipolar compressible Navier-Stokes-Poisson system with unequal viscosities
论文作者
论文摘要
给出了具有不平等粘度的3D可压缩双极纳维尔 - 斯托克斯 - 波森系统（BNSP）的时空行为。当推论BNSP的广义Huygens原理时，电场的时空估计是最重要的事情，因为只能通过Poisson方程从Poisson方程中获得$ \ nabla ϕ = \ frac {\ nabla}δ（ρ-n）$。因此，它需要证明$ρ-n $的时空估计仅包含扩散波。这些不平等系数的外观无法遵循特殊情况的想法，在这种情况下，原始系统被重写为可压缩的NS系统，并且在未知组合的线性组合后，将原始系统重写为可压缩的NS系统。这种线性组合为非线性术语带来了特殊的结构，并且该结构也用于获得$ρ-N $的所需时空估计。此外，在[36]中，Green的子系统NSP的功能不包含Huygens Wave。但是，对于一般情况下，这种线性组合的好处将不再存在。首先，我们必须直接考虑原始系统的$ 8 \ times8 $ green矩阵。其次，绿色功能中的所有条目在低频中实际上都包含波浪运算符。因此，这通常会为Green功能中的每个条目产生Huygens的波浪，因此，无法像往常一样实现$ρ-N $的时空估计。我们通过进行更详细的频谱分析并开发因Green功能中微妙的取消而产生的新估计来克服这一困难。第三，由于线性组合中非线性术语的特殊结构丢失，我们将开发新的非线性卷积估计值，以便我们最终可以获得$ \ nabla ϕ $的预期时空估计，并进一步验证广义的Huygens原理。
Space-time behaviors for 3D compressible bipolar Navier-Stokes-Poisson system (BNSP) with unequal viscosities are given. The space-time estimate of electric field $\nablaϕ$ is the most important thing when deducing generalized Huygens' principle for BNSP since this estimate only can be obtained by $\nablaϕ=\frac{\nabla}Δ(ρ-n)$ from the Poisson equation. Thus, it requires to prove that the space-time estimate of $ρ-n$ 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 $ρ-n$. 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 $ρ-n$ 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 $\nablaϕ$ and further verify the generalized Huygens' principle.
