A Blow-Up Criterion of Strong Solutions to the Quantum Hydrodynamic Model
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
A BLOW-UP CRITERION OF STRONG SOLUTIONS TO THE QUANTUM HYDRODYNAMIC MODEL∗
1É)
†
Guangwu WANG (
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China E-mail : [email protected]
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Boling GUO (
Institute of Applied Physics and Computational Mathematics, China Academy of Engineering Physics, Beijing 100088, China E-mail : [email protected] Abstract In this article, we focus on the short time strong solution to a compressible quantum hydrodynamic model. We establish a blow-up criterion about the solutions of the compressible quantum hydrodynamic model in terms of the gradient of the velocity, the second spacial derivative of the square root of the density, and the first order time derivative and first order spacial derivative of the square root of the density. Key words
Compressible quantum hydrodynamic model; blow-up criterion; strong solution
2010 MR Subject Classification
1
35A01; 35D35; 35M30; 35Q40
Introduction
In this article, we will investigate the blow-up criterion of the quantum hydrodynamic model(QHD): ∂t ρ + div(ρu) = 0,
√ ∆ ρ ∂t (ρu) + div(ρu ⊗ u) + ∇P (ρ) = ε2 ρ∇( √ ). ρ
(1.1) (1.2)
Here, ρ and u represent the density and the velocity of the fluid. And P (ρ) is the pressure. We choose the isentropic case for simplicity, for example: P (ρ) = ργ (γ > 1). The parameter ε is the scaled Planck constant. u ⊗ u denotes the matrix with element ui uj (i, j = 1, · · · , n). And √ ∆ ρ √ ρ is the quantum Bohm potential. In this article, we will consider that the initial data to equation (1.1)–(1.2) are ρ(x, 0) = ρ0 (x) > 0, u(x, 0) = u0 (x), ∗ Received
x ∈ Ω,
x ∈ Ω.
(1.3) (1.4)
January 14, 2019. The first author is supported by the National Natural Science Foundation of China (11801107); the second author is supported by the National Natural Science Foundation of China (11731014). † Corresponding author
796
ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
Here, Ω is the sub-domain of Rn . When we consider the boundary value problem, we can assume that the boundary conditions satisfy that √ u · ~n = 0, ∇ ρ · ~n = 0, x ∈ ∂Ω, (1.5) where ~n is the outward normal vector of ∂Ω. This model (1.1)–(1.2) was firstly derived by Madelung [28] in 1927 from the Schr¨odinger equation for the wave function ψ: ε2 ∆ψ + h(|ψ|2 )ψ, 2 ψ(x, 0) = ψ0 , x ∈ Rn , iε∂t ψ =
t > 0, x ∈ Rn ,
(1.6) (1.7)
by the so-called Madelung transformation iS √ (1.8) ρ exp ( ). ε Here, n ≥ 1, ε > 0 is the scaled Planck constant, and h is an integrable function such that ψ=
h′ (ρ) =
P ′ (ρ) . ρ
(1.9)
ρ = |ψ|2 is particle density, and S is a phase function. The Schr¨odinger equation (1.6) can be used to describe multi-particle approximations in the mean field theory of quantum mechanics, when one considers a large number of quantum particles acting in unison and takes into account only a finite number of particle-particle interactions. Next, we will give a brief derivation from the Schr¨odinger equa
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