Well-Posedness and Stability of Quantum Hydrodynamics for Semiconductors in ℝ³

Semiclassical and relaxation limits of bipolar quantum hydrodynamic model for semiconductors

“…That is, one can fix any of the two parameters ε and τ and let the other tend to zero. Moreover, our analysis for the bipolar model (1.10)-(1.12) can be applied to justify the semiclassical limit and relaxation limit for the unipolar model [20,14,15].…”

“…The framework used there is to study an extended problem derived based on a deposition of the original problem, which in turn implies the expected problem as a special case. The method employed in [26,15] can be applied to our bipolar model directly. The proof is straightforward, and we have Lemma 3.1 (Local existence) Let the parameters ε > 0, τ > 0, λ > 0 be fixed.…”

“…The well-posedness of steady state subsonic solutions has been proved also in [18,19,33]. Transient solutions are shown to exist either locally in time [13,14] or globally in time for data close to a steady state [15,16,19,26]. The bipolar hydrodynamic(HD) model of the global solutions has been studied in [9].…”

Relaxation Limit from the Quantum Navier–Stokes Equations to the Quantum Drift–Diffusion Equation

“…On a related subject, let us also comment on the inviscid counterpart of system (1.1), namely the quantum hydrodynamic (QHD) system Marcati 2009, 2016). Due to the dissipative term −ξρu, also in this case it is possible to show both the exponential convergence toward the stationary solution (Huang et al 2006(Huang et al , 2010 and the relaxation limit (Jüngel et al 2006), again toward the (QDD) equation. However the only available results here deal with small, regular perturbations around stationary solutions.…”

Large data solutions to the viscous quantum hydrodynamic model with barrier potential

“…Proving analytical results for is quite challenging, and one reason for that is the third‐order term in the second equation, where we have introduced the abbreviation for the so‐called Bohm potential. We mention results on local or global existence or various asymptotics (proved using energy methods) in . All these results omit the barrier potential V B .…”