Incompressible Fluid In Some Of The Mathematical Problem

The Navier-Stokes system is the most important system when.you study the fluid mechanics problems.It has been studied by P.-L.Lions,J.Y.Chemin,A.Majda and R.Danchin,and they did get a lot of wellposedness results of this system.Based on their work,we will study the Navier-Stokes-Coriolis system and the Kazhihov-Smagulov type model in this paper.The Navier-Stokes-Coriolis system is the model,which describes the ocean current and the Atmospheric circulation,and we mainly get the following results: In the Sobolev framework,we discuss the local wellposedness with different initial data.·In the Besov framework,we get the local wellposedness result for this system.·In the Sobolev framework,we get the convergence to the 2D Navier-Stokes system,asεtends to zero.Chemin has got the convergence of the densityindependent system to the 2D Navier-Stokes system when 2≤q≤6.And in this paper,we can extend the convergence results of the density-dependent system to the q＞6 situation.·In the framework of Sobolev spaces,we discuss the blow-up criterion of the smooth solutions,and further we can get the almost global wellposedness results.About the pollutant models of Kazhikhov-Smagulov type,we will talk about the following problems in this paper:·Local well-posedness for the Kazhikhov-Smagulov type equations withλ≥0.·Derivation of blow-up criteria. ·The small diffusion coefficient limit:A "weak result" The construction of local solutions combined with a result of convergence in L²-norm gives the existence of a fixed interval[0,T]on which the diffusion solution(ρλ,uλ,▽Πλ) tends strongly to the Euler solution whenλgoes to zero.·The small diffusion coefficient limit:A "strong result" If the Euler system has smooth solution on some given interval[0,T₀],then this system with smallλhas a solution on the same time interval.Besides, strong convergence holds true on[0,T₀]...