三维Stokes近似系统的局部强解

Local strong solutions of the three-dimensional Stokes approximation equations

目的研究三维Stokes近似系统在一个有界光滑区域上局部强解的存在性、唯一性、稳定性。方法通过一个半离散化的Galerkin格式构造逼近解,利用一致估计进而通过取极限来得到想要的解。结果假设初始值满足自然兼容性条件,本文允许初始密度真空的存在,即初始密度在Ω的一个开子集上为0,那么三维Stokes近似系统有局部强解。结论局部强解具有存在性、唯一性、稳定性。

Aim In order to study the existence, uniqueness, stability of the local strong solutions of the three-dimensional Stokes approximation equations in a bounded smooth domain. Methods The approximate solutions have been constructed via a semi-discrete Galerkin scheme, using uniform estimates and solutions have been obtained by passing to the limit. Results Provided that the initial data satisfies a natural compatibility condition, the initial density needs not be bounded away from zero. Namely, the initial density may vanish in an open subset of Ω, and then the three-dimensional Stokes approximation equations have a local strong solution. Conclusion The existence, uniqueness, stability of the local strong solutions is proved.