We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in R^n with compact and smooth boundary, subject to the kinematic and vorticity boundary conditi...We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in R^n with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat boundary. We observe that, under the nonhomogeneous boundary conditions, the pressure p can be still recovered by solving the Neumann problem for the Poisson equation. Then we establish the well-posedness of the unsteady Stokes equations and employ the solution to reduce our initial-boundary value problem into an initial-boundary value problem with absolute boundary conditions. Based on this, we first establish the well-posedness for an appropriate local linearized problem with the absolute boundary conditions and the initial condition （without the incompressibility condition）, which establishes a velocity mapping. Then we develop apriori estimates for the velocity mapping, especially involving the Sobolev norm for the time-derivative of the mapping to deal with the complicated boundary conditions, which leads to the existence of the fixed point of the mapping and the existence of solutions to our initial-boundary value problem. Finally, we establish that, when the viscosity coefficient tends zero, the strong solutions of the initial-boundary value problem in R^n（n ≥ 3） with nonhomogeneous vorticity boundary condition converge in L^2 to the corresponding Euler equations satisfying the kinematic condition.展开更多
基金supported in part by the National Science Foundation under Grants DMS-0807551, DMS-0720925, and DMS-0505473the Natural Science Foundationof China (10728101)supported in part by EPSRC grant EP/F029578/1
文摘We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in R^n with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat boundary. We observe that, under the nonhomogeneous boundary conditions, the pressure p can be still recovered by solving the Neumann problem for the Poisson equation. Then we establish the well-posedness of the unsteady Stokes equations and employ the solution to reduce our initial-boundary value problem into an initial-boundary value problem with absolute boundary conditions. Based on this, we first establish the well-posedness for an appropriate local linearized problem with the absolute boundary conditions and the initial condition （without the incompressibility condition）, which establishes a velocity mapping. Then we develop apriori estimates for the velocity mapping, especially involving the Sobolev norm for the time-derivative of the mapping to deal with the complicated boundary conditions, which leads to the existence of the fixed point of the mapping and the existence of solutions to our initial-boundary value problem. Finally, we establish that, when the viscosity coefficient tends zero, the strong solutions of the initial-boundary value problem in R^n（n ≥ 3） with nonhomogeneous vorticity boundary condition converge in L^2 to the corresponding Euler equations satisfying the kinematic condition.
文摘我国西北地区的水电与新能源打捆经电网换相高压直流(line-commutated converter HVDC,LCC-HVDC)输电系统送出,直流发生单极闭锁故障时送端系统将承受严重的过电压冲击。送端水电机组具有进相运行能力,可进一步发掘其动态无功容量用于抑制系统过电压,但水电机组进相协调的相关研究几近空白。此外,进相协调控制下水电机组的功角稳定特性尚不明确,亟需开展研究。为此,首先针对送端系统不同程度的过电压工况,提出水电机组进相协调策略,包括水电与直流换流站协调、水电机组间协调两部分;其次,基于电磁功率解析表达式和等面积准则(equal area criterion,EAC),研究直流单极闭锁前后水电机组的功角稳定特性;然后,以保障系统功角稳定为目标,分析直流单极闭锁前后水电机组最大进相深度的变化规律,并由此提出与进相协调策略相适配的低励限制曲线改进方法;最后,基于西北地区某实际风光水送端系统的PSCAD/EMTDC模型进行验证,结果表明:提出的水电机组进相协调策略及低励限制曲线改进方法,在保障功角稳定的前提下,进一步提高了送端系统抑制过电压能力,对实际工程具有一定指导意义。
