We consider the Cauchy problem, free boundary problem and piston problem for one-dimensional compressible Navier-Stokes equations with density-dependent viscosity. Using the reduction to absurdity method, we prove tha...We consider the Cauchy problem, free boundary problem and piston problem for one-dimensional compressible Navier-Stokes equations with density-dependent viscosity. Using the reduction to absurdity method, we prove that the weak solutions to these systems do not exhibit vacuum states, provided that no vacuum states are present initially. The essential re- quirements on the solutions are that the mass and energy of the fluid are locally integrable at each time, and the Lloc1-norm of the velocity gradient is locally integrable in time.展开更多
Like the progress made by Dirac that wave function ψ(x) was reformed as x |ψ,where x| is the coordinate representation,we endow the characteristic function χλ = Tr(e λa-λaρ) of density operator ρ with the mean...Like the progress made by Dirac that wave function ψ(x) was reformed as x |ψ,where x| is the coordinate representation,we endow the characteristic function χλ = Tr(e λa-λaρ) of density operator ρ with the meaning of wave function of |ρ in the thermal entangled state η| representation in the doubled Fock space,χλ = η = λ|ρ,where |ρ = ρ|η = 0.We find the time evolution of χλ can then be directly and neatly obtained via this approach.The way of deriving the density operator from η = λ | ρ is also presented.展开更多
The effects of Mg doping(MgAl) and native N vacancy(VN) on the electronic structures and transport properties of Al N nanowire(Al NNW) were theoretically investigated by using density functional theory. Either the MgA...The effects of Mg doping(MgAl) and native N vacancy(VN) on the electronic structures and transport properties of Al N nanowire(Al NNW) were theoretically investigated by using density functional theory. Either the MgAl defect or the VN defect prefers to be formed on the Al NNW surfaces. Both MgAl and VN defects could increase the conductivity owing to introducing a defect band inside the band gap of Al N and split the Al N band gap into two subgaps. The defect concentration has little influence on the magnitude of the subgaps. The MgAl serves as a shallow acceptor rendering the nanowire a p-type conductor. The VN introduces a deep donor state enabling the nanowire an n-type conductor. The MgAl systems exhibit higher conductivity than the VN ones owing to the narrow subgaps of MgAl systems. The conductivity is roughly proportional to the defect concentration in the MgAl and VN defect systems. When the MgAl and VN coexist, the hole state of the MgAl defect and the electron state of the VN defect will compensate each other and their coupling state appears just above the valence-band maximum leading to a little decrease of the band gap compared with the pure Al NNW, which is unfavorable for the enhancing of the conductivity.展开更多
基金Project supported by the National Natural Science Foundation of China (No. 10571158) and the DFG
文摘We consider the Cauchy problem, free boundary problem and piston problem for one-dimensional compressible Navier-Stokes equations with density-dependent viscosity. Using the reduction to absurdity method, we prove that the weak solutions to these systems do not exhibit vacuum states, provided that no vacuum states are present initially. The essential re- quirements on the solutions are that the mass and energy of the fluid are locally integrable at each time, and the Lloc1-norm of the velocity gradient is locally integrable in time.
基金supported by the National Natural Science Foundation of China (Grant No. 11175113)
文摘Like the progress made by Dirac that wave function ψ(x) was reformed as x |ψ,where x| is the coordinate representation,we endow the characteristic function χλ = Tr(e λa-λaρ) of density operator ρ with the meaning of wave function of |ρ in the thermal entangled state η| representation in the doubled Fock space,χλ = η = λ|ρ,where |ρ = ρ|η = 0.We find the time evolution of χλ can then be directly and neatly obtained via this approach.The way of deriving the density operator from η = λ | ρ is also presented.
基金supported by the National Natural Science Foundation of China(Grant Nos.51073048,51473042)the Leaders in Academe of Har-bin City of China(Grant No.2013RFXXJ024)the Science Foundation for Backup Leader of Leading Talent Echelon in Heilongjiang Province
文摘The effects of Mg doping(MgAl) and native N vacancy(VN) on the electronic structures and transport properties of Al N nanowire(Al NNW) were theoretically investigated by using density functional theory. Either the MgAl defect or the VN defect prefers to be formed on the Al NNW surfaces. Both MgAl and VN defects could increase the conductivity owing to introducing a defect band inside the band gap of Al N and split the Al N band gap into two subgaps. The defect concentration has little influence on the magnitude of the subgaps. The MgAl serves as a shallow acceptor rendering the nanowire a p-type conductor. The VN introduces a deep donor state enabling the nanowire an n-type conductor. The MgAl systems exhibit higher conductivity than the VN ones owing to the narrow subgaps of MgAl systems. The conductivity is roughly proportional to the defect concentration in the MgAl and VN defect systems. When the MgAl and VN coexist, the hole state of the MgAl defect and the electron state of the VN defect will compensate each other and their coupling state appears just above the valence-band maximum leading to a little decrease of the band gap compared with the pure Al NNW, which is unfavorable for the enhancing of the conductivity.
