Blowup Mechanism for a Fluid-Particle Interaction System in R 3 $\mathbb{R}^{3}$
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Blowup Mechanism for a Fluid-Particle Interaction System in R3 Jinrui Huang1 · Bingyuan Huang2 · Yuqin Wu3
Received: 12 October 2019 / Accepted: 13 April 2020 © Springer Nature B.V. 2020
Abstract We study the Cauchy problem and the mixed initial boundary value problem of a fluid-particle interaction system in R3 . A Serrin type criterion for the strong solution of the ∞ and uLs Lr , where 2/s + 3/r ≤ 1 and Cauchy problem is established in terms of ρL∞ t Lx t x 3 < r ≤ ∞. In view of some useful integral inequalities, we prove the life span estimates of the regular solution. Keywords Compressible · Fluid-particle interaction model · Serrin type criterion · The life span Mathematics Subject Classification 35Q35 · 76N10
1 Introduction In this manuscript, we study the following interaction system, which is called the compressible Navier-Stokes-Smoluchowshi model (see Carrillo-Goudon [4]): ρt + div(ρu) = 0,
(1.1)
ρut + ρu · ∇u + ∇(P + η) = μu + λ∇ div u − (βρ + η)∇,
(1.2)
ηt + div(ηu) = η + div(η∇),
(1.3)
B B. Huang
[email protected] J. Huang [email protected] Y. Wu [email protected]
1
School of Mathematics and Computational Science, Wuyi University, Jiangmen 529020, P.R. China
2
School of Mathematics and Statistics, Hanshan Normal University, Chaozhou 521041, P.R. China
3
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, P.R. China
J. Huang et al.
where x ∈ ⊂ R3 and t ≥ 0, ρ, u = (u1 , u2 , u3 ) and η are functions representing the density of the fluid, the fluid velocity field, and the density of the particles in the mixture, respectively. The pressure function P = aρ γ , a > 0, γ > 1. The external potential = (x) : → R+ describes the influence of gravity and buoyant force, β is a generic constant describing the different effects on the fluid and the particles arisen from the external force, the constant viscosity coefficients λ and μ possess the constrains: μ > 0, λ + 23 μ ≥ 0. The Navier-Stokes-Smoluchowshi system (1.1)-(1.3) is prescribed with the initial condition (ρ, u, η)|t=0 = (ρ0 (x), u0 (x), η0 (x)), x ∈ ,
(1.4)
and one of two boundary conditions as follows: Case one: if = R3 , we consider the far-field condition with (ρ, u, η) → (0, 0, 0), as |x| → ∞.
(1.5)
Case two: if = R3+ is the half space, we require the condition that u · n|∂ = −u3 |∂ = 0, n × (∇ × u)|∂ = ∂3 u1 − ∂1 u3 , ∂3 u2 − ∂2 u3 , 0 = 0,
(1.6)
(∇η + η∇) · n|∂ = − (∂3 η + η∂3 ) |∂ = 0, with n = (0, 0, −1) representing the outer normal vector to ∂. Case two involves some boundary effect which naturally will introduce some analytic challenge.
1.1 Research Background There are many relevant papers on system (1.1)-(1.3) in recent years. In one dimension, the authors in [10, 20] established the well-posedness of the global regular solutions. In dimension three, Ballew-Trivisa in [2] studied the global existence of suitable weak solutions, and soon later, they established the existence of weakly dissipative solutions in [3]. The readers are also referred to a rela
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