Global existence and time decay rates for the 3D bipolar compressible Navier-Stokes-Poisson system with unequal viscosit
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https://doi.org/10.1007/s11425-020-1719-9
Global existence and time decay rates for the 3D bipolar compressible Navier-Stokes-Poisson system with unequal viscosities Guochun Wu1 , Yinghui Zhang2,∗ & Anzhen Zhang1 1School 2School
of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China; of Mathematics and Statistics, Guangxi Normal University, Guilin 541004, China
Email: [email protected], [email protected], [email protected] Received March 20, 2020; accepted June 18, 2020
Abstract
We consider the Cauchy problem for the three-dimensional bipolar compressible Navier-Stokes-
Poisson system with unequal viscosities. Under the assumption that the H 3 norm of the initial data is small but its higher order derivatives can be arbitrarily large, the global existence and uniqueness of smooth solutions are −s obtained by an ingenious energy method. Moreover, if additionally, the H˙ −s ( 21 6 s < 32 ) or B˙ 2,∞ ( 21 < s 6 32 ) norm of the initial data is small, the optimal decay rates of solutions are also established by a regularity interpolation trick and delicate energy methods. Keywords MSC(2010)
bipolar, Navier-Stokes-Poisson, global existence, optimal decay rates 76W05, 35Q35, 35D05
Citation: Wu G C, Zhang Y H, Zhang A Z. Global existence and time decay rates for the 3D bipolar compressible Navier-Stokes-Poisson system with unequal viscosities. Sci China Math, 2021, 64, https://doi.org/10.1007/s11425-020-1719-9
1
Introduction
At high temperature and velocity, ions and electrons in a plasma tend to become two separate fluids due to their different physical properties (inertia, charge). The bipolar Navier-Stokes-Poisson (BNSP) system in plasma physics describes dynamics of two separate compressible fluids of ions and electrons interacting with their self-consistent electromagnetic field. In the present paper, we are concerned with the following 3D bipolar Navier-Stokes-Poisson system which describes a plasma composed of electrons and one species of ions: ∂t ni + div(ni ui ) = 0, m ∂ (n u ) + mi div(ni ui ⊗ ui ) + ∇Pi = µi ∆ui + λi ∇divui + Zni e∇ϕ, i t i i ∂t ne + div(ne ue ) = 0, (1.1) me ∂t (ne ue ) + me div(ne ue ⊗ ue ) + ∇Pe = µe ∆ue + λe ∇divue − ne e∇ϕ, lim ϕ = 0 ∆ϕ = 4πe(Zni − ne ), |x|→∞
* Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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with the initial conditions
(
(ni , ui , ne , ue , ϕ)(x, 0) = (ni,0 , ui,0 , ne,0 , ue,0 , ϕ0 ) →
1 , 0, 1, 0, 0 Z
) as |x| → ∞
(1.2)
for (x, t) ∈ R3 × R+ . The above symbols mean that the ions have the density ni , mass mi , velocity ui , pressure Pi = Pi (ni ) and charge Ze, and the electrons have the density ne , mass me , velocity ue , pressure Pe = Pe (ne ) and charge −e, where Z, mi and me are positive constants. The viscosity coefficients satisfy the usual physical conditions, i.e., µi > 0,
3µi + 2λi > 0,
µe > 0,
3µe + 2λe > 0.
Due to the physic
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