Abstract
In this paper, we are concerned with the existence of global weak solutions to the compressible Navier–Stokes–Poisson equations with the non-flat doping profile when the viscosity coefficients are density-dependent, the data are large and spherically symmetric, and we focus on the case where those coefficients vanish in vacuum. We construct a suitable approximate system and consider it in annular regions between two balls. The global solutions are obtained as limits of such approximate solutions. Our proofs are mainly based on the energy and entropy estimates.
Highlights
The dynamics of charged particles of one carrier type can be described by the compressible Navier–Stokes–Poisson equations:⎧ ⎪⎪⎨ρt + div(ρU) = 0, ⎪⎪⎩(ρΦU )t = + ρ div(ρ – b, U ⊗ ) + ∇ P(ρ div(h(ρ )D(U
By the a priori estimates established in Lemmas 4.1–4.5 for and a continuity argument, we show that it is a global classical solution to (4.16) with the initial data (ρ0δ, uδ0) and the boundary condition (5.4)
We prove that the right-hand side of (5.76) tends to zero as j → ∞
Summary
The dynamics of charged particles of one carrier type (e.g., electrons) can be described by the compressible Navier–Stokes–Poisson equations:. Degeneration at vacuum occurs because of the dependence of viscosity coefficients on flow density, which makes it very difficult to derive a uniform a priori estimate for the velocity and trace the motion of particle paths near vacuum regions It is not known yet whether or not the vacuum states form for global (weak) solutions to (1.4) even if initial data is far from vacuum. By taking a limit, we show that a global spherically symmetric entropy weak solution to (1.1) exists for general initial data with finite entropy for γ ∈ (1, 3) It is different from the situation in [10] where the two viscosity coefficients need to satisfy the relation μ(ρ) = μ1Ψ (ρ), λ(ρ) = 2μ1(ρΨ (ρ) – Ψ (ρ)), and μ(ρ) has to satisfy another condition μ(ρ). T , u0 L4 , ρ04 x L4 , and so the lemma is proved
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