The Free Boundary Problems And Low Mach Number Limit Of Several Types Of Fluid Equations

Fluid mechanics is an important branch of mechanics.It mainly researches the motion laws of fluids under the action of various forces.We commit to explaining some related physical phenomena from a mathematical point of view,which has some guiding effects on daily applications.In this thesis,there mainly contains two kinds of problems.On t.he one hand,we consider the global well-posedness of the magnetic field with free boundary,provided the small initial data.On the other hand,we research the low Mach number limit problem for two kinds of compressible fluids in a bounded domain with the velocity satisfies the Dirichlet boundary condition.In Chapter 1,we introduce the physica.l background of the matheinatical models we researched,and introduce t.he physical meaning of studying the low Mach number limit problems and the free boundary problems.Finally,we present the research status of the problems we considered and the main results of this thesis.In Chapter 2,we consider a layer of a viscous incompressible electrically conduct-ing fluid int.eracting with the magnetic filed in a horizontally periodic setting.The upper boundary bounded by a free boundary and below bounded by a flat rigid inter-face.We prove the global well-posedness of the problem for both the case with and without surface tension.Moreover,we show that t.he global solution decays to the equilibrium exponentially in the case with surface tension,however the global solution decays to the equilibrium at an almost,exponential rate in the case wit.hout surface t.ension.In Chapter 3.we consider t.he low Mach mumber limit of the full compressible MHD equations in a 3-D bounded domain with Dirichlet boundary condition for ve-locity field,Neumann boundary condition for temperatuie nd porfectiv conducti boundary condition for magnetic First.the uniform estimates in the Mach mun-ber for the strong solutions aro obtianed in a short time interval.provided that the initial density and temperature are close to the constant states.Then.we prove the solutions of the full compressible MHD equations converge to the isentropic incom-pressible MHD equations as the Mach number tends to zero.In Chapter 4,we consider the uniform estimates of strong solutions in the Mach number ? and t ?[0.?)for the compressible nematic liquid crystal flows in a 3D bounded domain ?(?)R3,provided the initial data are small enough and the density is close to the constant state.Here we consider the case that the velocity field satisfies the Dirichlet boundary condition.Bayed on the uniform estimates,we obtain the global convergence of the solution for the compressible system to the incompressible system as the Mach number tends to zero.In Chapter 5.we make a summary to our thesis and have a plan for the future work.