Two Models Including Transport Equation In Meteorology And Semiconductors

The semi-geostrophic model and the quantum drift-diffusion model are investi-gated in this thesis, which are used to describe frontogenesis in meteorology and thequantum effects of semiconductor devices respectively. Both of them are composed ofa mass conserving transport equation and an elliptic equation.Firstly, we study the radial solutions of Monge-Ampe`re equations and the semi-geostrophic model. By bootstrap method, we obtain an (optimal) regularity conditionfor radial solutions of the most general real Monge-Ampe`re equations, which completesthe radial solution theory of Monge-Ampe`re equations and is believed to be useful inthe research of fully nonlinear elliptic equations. By using a part of above conclusion,we discuss the traveling wave radial solutions of semi-geostrophic model, which isa favorable attempt in the search for solutions with better regularity than the knownmeasure solution.Secondly, we systematically investigate the existence, semiclassical limit andlong-time behavior of weak solutions to the quantum drift-diffusion model. Makinguse of an exponential transformation technique and entropy estimate method, we getthe global existence and semiclassical limit of nonnegative weak solutions to one-dimensional unipolar transient isentropic model with nonnegative initial and homoge-nous Neumann (or periodic) boundary conditions. At the same time, by using a log-arithmic Sobolev inequality, an exponential time decay estimate of the periodic weaksolution to above (or isothermal ) model is obtained, which shows that the solutionexponentially approaches its mean value as time increases.