The Existence and Long-Time Behavior of Weak Solution to Bipolar Quantum Drift-Diffusion Model*

Abstract

The authors study the existence and long-time behavior of weak solutions to the bipolar transient quantum drift-diffusion model, a fourth order parabolic system. Using semi-discretization in time and entropy estimate, the authors get the global existence of nonnegative weak solutions to the one-dimensional model with nonnegative initial and homogenous Neumann (or periodic) boundary conditions. Furthermore, by a logarithmic Sobolev inequality, it is proved that the periodic weak solution exponentially approaches its mean value as time increases to infinity.

Viscous quantum hydrodynamics and parameter-elliptic systems

“…First we derive a resolvent estimate for A, improving the a priori estimates of (6). Let ∈ ϑ with | | 0 (p) as in Theorem 2.1, and consider the problem…”

Lower bound of decay rate for higher order derivatives of solution to the compressible quantum magnetohydrodynamic model

“…We also remark that Chen and Dreher 14 proved the local existence of solution to the viscous model of QHDs in , and they showed the local existence of solution in higher dimensions under the periodical boundary condition. For more results about the well‐posedness of solutions to the quantum fluid model, readers can refer to previous studies 15‐21 and the references therein.…”