Blow-up of Smooth Solution to the Isentropic Compressible Navier-Stokes-Poisson System with Compact Density
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Blow-up of Smooth Solution to the Isentropic Compressible Navier-Stokes-Poisson System with Compact Density Baoquan Yuan1
· Xiaokui Zhao2
Received: 21 November 2014 / Accepted: 4 September 2017 / Published online: 9 October 2017 © Springer Science+Business Media B.V. 2017
Abstract The blow-up of smooth solution to the isentropic compressible Navier-StokesPoisson (NSP) system on Rd is studied in this paper. We obtain that if the initial density is compactly supported, the spherically symmetric smooth solution to the NSP system on Rd (d ≥ 2) blows up in finite time. In the case d = 1, if 2μ + λ > 0, then the NSP system only exits a zero smooth solution on R for the compactly supported initial density. Keywords Compressible Navier-Stokes-Poisson system · Smooth solution · Blow-up Mathematics Subject Classification (2000) 76W05 · 35B65 · 35B44
1 Introduction We consider the following compressible Navier-Stokes-Poisson (NSP) system on Rd , d ≥ 1, ⎧ ⎪ ⎨∂t ρ + div(ρu) = 0, ∂t (ρu) + div(ρu ⊗ u) + ∇p = μu + (μ + λ)∇ div u + ρ∇Φ, ⎪ ⎩ Φ = ρ,
(1.1)
which is used to model and simulate the transport of charged particles in semiconductor devices, see Ref. [5]. Here ρ, p = ρ γ , u = (u1 , u2 , . . . , ud )T , Φ are the unknown density, pressure, velocity and electrostatic potential respectively. γ > 1 is the ratio of specific heat,
B B. Yuan
[email protected] X. Zhao [email protected]
1
School of Mathematics and Information Science, Henan Polytechnic University, Henan, 454000, China
2
School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China
190
B. Yuan, X. Zhao
λ and μ are, respectively, the shear viscosity coefficient and the bulk viscosity coefficient satisfying the conditions: μ > 0,
2μ + dλ 0.
(1.2)
The NSP system is supplemented with the initial data (ρ, u, Φ)|t=0 = (ρ0 , u0 , Φ0 ) ∈ W m,1 Rd
(1.3)
with m ≥ d + 2. We will assume that the initial density ρ0 (x) has compact support so that there exists a positive constant R0 such that supp ρ0 (x) ⊂ BR0 , where BR0 {x ∈ Rd ||x| R0 } is a ball. Without the potential Φ, the isentropic and full compressible NSP systems, respectively, reduce to the corresponding compressible Navier-Stokes equations
∂t ρ + div(ρu) = 0, ∂t (ρu) + div(ρu ⊗ u) + ∇P = μu + (μ + λ)∇ div u,
(1.4)
and the full compressible Navier-Stokes equations ⎧ ⎪ ⎨∂t ρ + div(ρu) = 0, ∂t (ρu) + div(ρu ⊗ u) + ∇P = μu + (μ + λ)∇ div u, ⎪ 2 ⎩ ∂t (ρe) + div(ρue) − κe + P div u = μ2 ∇u + ∇uT + λ(div u)2 ,
(1.5)
where e is the unknown specific internal energy, and κ is the thermal conductivity coefficient. For the full compressible Navier-Stokes equations (1.5), it was first proved by Z.P. Xin [9] that the non-zero smooth solution blows up in finite time if κ = 0 and the initial density of (1.5) is compactly supported on Rd (d ≥ 1). Xin-Yan [10] improved the blow-up results of [9] on Rd (d ≥ 2) by removing the assumptions that the initial density has compact support and the smooth solution has finite energy, but the initial data only have an isolated mass group. In t
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