Optimal decay rate of the bipolar Euler–Poisson system with damping in dimension three

Wu, Zhigang; Qin, Yuming

doi:10.1002/mma.3269

Cited by 3 publications

(1 citation statement)

References 39 publications

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“…For the compressible Euler-Poisson system and related models, we refer to [14,15,16,31,32] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

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

Chen,

Wu,

Zhang

2021

Preprint

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This paper is concerned with the Cauchy problem of the 3D compressible bipolar Navier-Stokes-Poisson (BNSP) system with unequal viscosities, and our main purpose is threefold: First, under the assumption that H l ∩ L 1 (l ≥ 3)-norm of the initial data is small, we prove the optimal time decay rates of the solution as well as its all-order spatial derivatives from oneorder to the highest-order, which are the same as those of the compressible Navier-Stokes equations and the heat equation. Second, for well-chosen initial data, we also show the lower bounds on the decay rates. Therefore, our time decay rates are optimal. Third, we give the explicit influences of the electric field on the qualitative behaviors of solutions, which are totally new as compared to the results for the compressible unipolar Navier-Stokes-Poisson(UNSP) system [Li et al., in

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“…For the compressible Euler-Poisson system and related models, we refer to [14,15,16,31,32] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

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

Chen,

Wu,

Zhang

2021

Preprint

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Decay estimates of solutions to the bipolar compressible Euler–Poisson system in $$\pmb {\mathbb {R}^3}$$

Tong

Tan

2020

Z. Angew. Math. Phys.

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

Zhang

2020

Sci. China Math.

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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.

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Optimal decay rate of the bipolar Euler–Poisson system with damping in dimension three

Cited by 3 publications

References 39 publications

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

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

Decay estimates of solutions to the bipolar compressible Euler–Poisson system in $$\pmb {\mathbb {R}^3}$$

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

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