The Well-posedness Of Geometric Dispersive Type Equations

In this thesis,we consider the geometric dispersive equations and their applications.Nonlinear dispersive equations arise in phenomena of wave propagation coming from physics and engineering such as water waves,optics,lasers,ferromagnetism,particle physics,general relativity and many others.Schrodinger map flow and wave map equation are the most typical models in geometric dispersive pde’s,where Schrodinger map flow is an interesting equation from physics known as the equation of ferromagnetic chain,and wave map equations are known as σ-model in physics literature and also are connected with certain special cases of Einstein’s equation of gravity.Other examples of geometric dispersive equations include the Maxwell-Klein-Gordon equation,the hyperbolic Yang-Mills equation,the hyperbolic minimal surface equation,the Landau-Lifshitz flow and so on.It is a key problem how the geometries of backgroud manifold and target manifold affect the long time behavior of the flow.There have been a lot of works on geometric dispersive equations.In fact,Tataru,Tao,Kenig,Merle,Klainerman and so on have made outstanding contributions in this field.It becomes an important direction in the field of dispersive equations.In this thesis,we consider equivariant Schrodinger flow of maps from H2 to S2 and 3-D hyperbolic Ericksen-Leslie’s liquid crystal model,and prove the small data global regularity.First,we consider the Schrodinger flow of maps from two dimensional hyperbolic space H2 to sphere S2,and prove the local well-posedness for large data and global well-posedness of equivariant Schrodinger flow with small data.Here we prove the local existence and uniqueness of Schrodinger flow for initial data using an approximation scheme and parallel transport introduced by McGahagan[64].Next,using the Coulomb gauge,we reduce the study of the equivariant Schrodinger flow to that of a system of coupled Schrodinger equations with potentials.Then we prove the global existence of equivariant Schrodinger flow for small initial data by Strichartz estimates and perturbation method.Second,hyperbolic Ericksen-Leslie’s liquid crystal model is a nonlinear coupling of Navier-Stokes equations with wave map to S2,which was established by Ericksen and Leslie during the period of 1958 through 1968,including the incompressible model and compressible model.The study of hyperbolic Ericksen-Leslie’ s system is at the infancy compared with its parabolic analogue.Here,we prove the small data global regularity for 3-D incompressible Ericksen-Leslie’s hyperbolic model without kinematic transport and simplified compressible Ericksen-Leslie’s hyperbolic model.Since they are quasilinear systems,our argument is a combination of vector field method and Fourier analysis.The main strategy to prove global regularity relies on an interplay between the control of high order energies and decay estimates,which is based on the idea inspired by the method of space-time resonances.The space-time resonance,developed by Germain,Masmoudi and Shatah[21-23],has been applied in quadratic Schrodinger equations,Euler-Maxwell equations,water waves etc.In order for the decay estimates,for incompressible hyperbolic liquid crystal model without kinematic transport,we use the dissipation to prove the decay;for simplified compressible hyperbolic liquid crystal model,we investigate the wave propagation of low-frequency part and the dissipation of high-frequency part to obtain the decay.Next,in order for the energy estimates,although the presence of time resonance without matching null structure,we ultilize the property that the space resonance set is empty and the dissipation to prove that the energy increse slowly.Hence,we get the small data global regularity.