Global solutions and relaxation limit to the Cauchy problem of a hydrodynamic model for semiconductors

Abstract

In this paper, we study the Cauchy problem for the one-dimensional Euler-Poisson (or hydrodynamic) model for semiconductors, where the energy equation is replaced by a pressure-density relation. First, the existence of global entropy solutions is proved by using the vanishing artificial viscosity method, where, a special flux approximate is introduced to ensure the uniform boundedness of the electric field E and the a-priori L∞ estimate, 0<2δ≤ρε,δ≤M(t),|uε,δ|≤M(t), where M(t) could tend to infinity as the time t tends to infinity, on the viscosity-flux approximate solutions (ρε,δ,uε,δ); Second, the compensated compactness theory is applied to prove the pointwise convergence of (ρε,δ,uε,δ) as ε,δ go to zero, and that the limit (ρ(x,t),u(x,t)) is a global entropy solution; Third, a technique, to apply the maximum principle to the combination of the Riemann invariants and ∫−∞xρε,δ(x,t)−2δdx, deduces the uniform L∞ estimate, 0<2δ≤ρε,δ≤M,|uε,δ|≤M, independent of the time t and ε,δ; Finally, as a by-product, the known compactness framework [14,29] is applied to show the relaxation limit, as the relation time τ and ε,δ go to zero, for general pressure P(ρ).