Some Mathematical Theories On Ericksen-Leslie Liquid Crystal Model And Navier-Stokes-Maxwell Model

As two typical models in the complex fluid,the compressible Ericksen-Leslie' s hyperbolic liquid crystal model and the Navier-Stokes-Maxwell electromagnetic model describe the dynamic behavior of the liquid crystal molecules in the compressible fluid and the motion of the conducting particles in an electromagnetic field,respectively.The first part of the thesis is cconcerned with the well-posedness of the compressible hyperbolic Ericksen-Leslie liquid crystal model,and the existence of its limit when the fluid velocity is given,meanwhile,the limit problem connecting a scaled wave map with the heat flow.As the second part of the thesis,the existence of the.global-in-time solution and its limit for the scaled Navier-Stokes-Maxwell equations with Ohm law is considered.The thesis can be divided into the following six chapters.In Chapter 1 we present the corresponding physical and mathematical backgrounds of the prolblems we considered and the ma.in results obtained in this thesis.Chapter 2 is concerned with the well-posedness of the Ericksen-Leslie's parabolic-hyperbolic liquid crystal model in compressible flow.Inspired by the study for incorm-pressible case[40]and some techniques from compressible Navier-Stokes equations,we prove the local-in-t.ime existence of the classical solution to the system with finite initial energy,under some constraints on the Leslie coefficients which ensure the basic energy law is dissipative.Furthermore,with an additional assumption on the coefficients which provides a damping effect,and the smallness of the initial cenergy,the global classical solution can be established.Cha.pter 3 deals with the limit of the hyperbolic Ericksen-Leslie liquid crystal model when the fluid veloeity is given.Formally,when the inertia constant goes to zero,the hyperbolie system of Ericksen-Leslie's liquid erystal flow reduces to the corresponding parabolic system.Under the assumptions that the initial data are well-prepared and the fluid velocity is given,we justify this limit in the context of classical soluticons.In Chapter 4 we study a limit connecting a scaled wave map with the heat flow into the unit sphere S2.We.show quantitatively how that the two equations are connected by means of an initial layer correction.This limit is motivated as a first step into un-derstanding the limit of zero inertia for the hyperbolic-parabolic Ericksen-Leslie's liquid crystal model.Chapter 5 is devoted to the limit of the Navier-Stokes-Maxwell system with Ohm 's law.The two-fluid incompressible Navier-Stokes-Maxwell system with solenoidal Ohm's law can be viewed as an asymptotic regime of incompressible Navier-Stokes-Maxwell system with Ohm's law.This was formally derived in Arsenio-Saint-Raymond's work[2].We justify rigorously this limit in the context of global-in-time classical solutions.The key is to derive the global-in-time uniform in ? energy estimate of the rescaled system with Ohm's law by employing the decay properties of both the electric field E? and the wave equation with linear damping of the divergence free magnetic field B?,then take the limit as ??0 to obtain the solutions of the system with solenoidal Ohm's law.In Chapter 6,we conclude this thesis by discussing some related mathematical prob-lems what we can do in the future.