Several Studies On The Existence Of Quantum Fluid Equations And The Non-viscous Limits Of Fluids

Quantum mechanic is a very important branch of the modern physics,which is focus on the motion of microscopic particles.Quantum hydrodynamic models can be used to describe many physical phenomena such as superconductor,superfluid.Bose-Einstein condensate,semiconductor.First of all,we introduce some models in fluid mechanic,provide the physical background and research status in Chapter 1.In Chapter 2,we will focus on the blow-up of the smooth solutions to several kinds of the quantum hydrodynamic models.In Section 2.1,we firstly prove the local-in-time existence of the smooth solution to initial problem of the quantum hydrodynamic model(quantum Euler equation)in Rd(d ≥ 1).Then we will show the blow-up of these smooth solutions.We research the blow-up of the smooth solution to the quantum hydrodynamic model with the homogeneous slip boundary condition in the half space in Section 2.2.In the last section we get the blow-up of the smooth solutions to four kinds of the viscous quantum hydrodynamic models.Ferromagnetic fluid equation is usually to describe the dissipative theory of the ferromagnet.In Chapter 3,we prove the global existence of the finite energy weak solution to the viscous quantum Navier-Stokes-Landau-Lifshitz-Maxwell equation in T2.Inviscid limit of the viscous fluid is a very important problem in fluid mechan-ic.In Chapter 4,we establish the vanishing viscosity limit about the incompressible magnetohydrodynamic equation with the generalized slip boundary condition.In Section 5.1,we prove that the smooth solution to the initial problem in Rn of the full compressible Navier-Stokes equation or isentropic compressible Navier-Stokes equation will blow up in the finite time.These results also hold for the initial boundary value problem in R+n.The Ericksen-Leslie equation can be used to describe the motion of crystal molecule of the nematic liquid crystal.In Section 5.2,we investigate the blow-up of the smooth solution to the compressible Ericksen-Leslie equation.We consider the initial problem in Rn,initial boundary value problem in R+n or unit sphere respectively.We prove the existence and uniqueness of the inhomogeneous incompressible Navier-Stokes-Landau-Lifshitz equation in two dimension in Chapter 6.Finally,we make a summary to our thesis,and make a prospect to the future in Chapter 7.