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

Abstract

We discuss analytically the stationary viscous quantum hydrodynamic model including a barrier potential, which is a nonlinear system of partial differential equations of mixed order in the sense of Douglis–Nirenberg. Combining a reformulation by means of an adjusted Fermi level, a variational functional, and a fixed point problem, we prove the existence of a weak solution. There are no assumptions on the size of the given data or their variation. We also provide various estimates of the solution that are independent of the quantum parameters. Copyright © 2015 John Wiley & Sons, Ltd.

The combined viscous semi-classical limit for a quantum hydrodynamic system with barrier potential

“…The maximum cannot be attained on the left endpoint s 0 , by assumption. If the maximum is attained at the right endpoint, then (9) and the Lipschitz continuity of u 0 imply max J κ u 2 κ (x) ≤ u 2 0 (s 0 +) + Cκ 1/4 . And if the maximum is attained inside of J κ , then (40) and (36) yield…”

“…Although formulated for the isothermal case p(n) = (T 0 + μ)n, the results of [12] seem to generalize to the case of general pressure terms p(n). And we also mention [9], where it was shown (in the isothermal case) that solutions (n, V, J) ∈ W 2,2 (0, 1) × W 2,2 (0, 1) × W 1,2 (0, 1) to (2) for given (possibly large) Dirichlet boundary values for V and periodic boundary values for n do exist. The purpose of the present paper is to extend the solution theory of [9] towards an asymptotic expansion of the solution, for vanishing values of the quantum mechanical parameters ε and ν, focusing on the equilibrium case.…”

“…We also assume the physically reasonable situation where V B + V κ vanishes at the endpoints of the interval [0, 1]. By a straightforward generalisation of the approach of [9], the known results read as follows, formulated in terms of u := √ n: [9,20].) Let ε, λ, ν, τ > 0, suppose C, V B ∈ L ∞ (0, 1), and assume that the Fermi level is a constant function F ∈ R. Let h be the enthalpy to the smooth and strictly monotonically increasing pressure term p, which fulfils sh (s) = p (s), s > 0, and additionally assume…”

“…The solution theory in [9] is based on a reformulation of the system (2) by means of a viscosity-adjusted Fermi level…”