A Sixth-Order Nonlinear Parabolic Equation for Quantum Systems

Abstract

Abstract. The global-in-time existence of weak nonnegative solutions to a sixth-order nonlinear parabolic equation in one space dimension with periodic boundary conditions is proved. The equation arises from an approximation of the quantum drift-diffusion model for semiconductors and describes the evolution of the electron density in the semiconductor crystal. The existence result is based on two techniques. First, the equation is reformulated in terms of exponential and power variables, which allows for the proof of nonnegativity of solutions. The existence of solutions to an approximate equation is shown by fixed-point arguments. Second, a priori bounds uniformly in the approximation parameters are derived from the algorithmic entropy construction method which translates systematic integration by parts into polynomial decision problems. The a priori estimates are employed to show the exponential time decay of the solution to the constant steady state in the L 1 norm with an explicit decay rate. Furthermore, some numerical examples are presented.Key words. Sixth-order parabolic equation, quantum semiconductors, global existence of solutions, algorithmic entropy construction, convergence of transport plans, long-time behavior of the solutions.

Entropies for radially symmetric higher-order nonlinear diffusion equations

“…The Lyapunov property of E α for α = 1 and d = 1 is proved in [13]. The proof of this property for α = 1 and d > 1 as well as the entropy production inequality are new.…”

“…Entropy estimates for the sixth-order quantum diffusion model (1.3) with periodic boundary conditions are available only in one space dimension. In fact, it has been shown in [13] that E 1 is an entropy and (1.4) holds for some c > 0 and with…”

“…When the non-local quantum diffusion model is expanded to order 4 , the main part of the differential operator is of sixth order, and the corresponding equation reads as The one-dimensional version of (1.3) has been derived in [11]; see Appendix A for the derivation in the general case. The one-dimensional equations with periodic boundary conditions have been analyzed in [13].…”

Physical and numerical viscosity for quantum hydrodynamics

“…In fact, we expect that the electrons cool down dramatically as they penetrate the potential barriers such that the isothermal model is not accurate enough. It is interesting to observe that there are NDR regions also at room temperature; this is not the case of the quantum drift-diffusion model [27]. In Figure 4.3 the currentvoltage curves for various values of the effective mass constant α with m eff = α · m 0 are shown.…”

“…The peak-to-valley ratios are given in Table 4.2. Interestingly, the peakto-valley ratios are not monotoneous with the effective mass like in the quantum drift-diffusion model [27]. temperature T 0 = 77 K. In order to obtain NDR effects, we need to choose a smaller relaxation time than that taken in the isothermal model.…”