三维不等粘性双极可压缩Navier-Stokes-Poisson方程组的点态时空估计
Pointwise space-time estimates of 3D bipolar compressible Navier-Stokes-Poisson system with unequal viscosities
摘要：
给出了三维可压缩不等粘性双极Navier-Stokes-Poisson系统的时空行为。在推导BNSP的广义惠更斯原理时，电场的时空估计$\nablaϕ$是最重要的，因为这种估计只能由$\nablaϕ=\frac{\nabla}Δ(ρ-n)$从泊松方程得到。因此，需要证明$ρ-n$的时空估计只包含扩散波。这些不相等的系数的出现导致人们不能遵循特殊情况下的想法，其中原始系统在未知量的线性组合之后被重写为可压缩NS系统和可压缩（单极）NSP系统。这种线性组合为非线性项带来了特殊的结构，这种结构也被用来得到$ρ-n$的理想时空估计。此外，不含惠更斯波的子系统NSP的格林函数在[36]中同样重要。然而，对于一般情况，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，低频格林函数中的所有项实际上都包含波算子。这通常会对格林函数中的每一项产生惠更斯波，因此，人们不能实现$ρ-n$的时空估计像往常一样只包含扩散波。我们克服了这个困难，采取更详细的谱分析和开发新的估计所产生的微妙的取消格林函数。第三，由于线性组合中非线性项的特殊结构丢失，我们将发展新的非线性卷积估计，从而最终得到$\nablaϕ$的期望时空估计，进一步验证广义惠更斯原理。
英文摘要：
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.
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