The Well-posedness And Asympototic Limit Of Some Quantum Hydrodynamic Models And Related Models

In this thesis,we are devoted to the studies of the well-posedness and asympototic limit of some quantum hydrodynamic models and related models.Studies on the fluid mechanics models can provide a theoretical basis for the application of material science,aerospace,nuclear,microelectronic technology and modern physical.This paper in this paper consists of seven parts.In Chapter 1,introduction.Firstly,we introduce the physical background and show the relevant models and their research progress.Then,we give the structure and the symbols of this thesis.In Chapter 2,we establish the quasineutral limit for the compressible quantum Navier-Stokes-Poisson equations with heat conductivity in the whole space 3?.We prove,it will lead to an incompressible Navier-Stokes equation as Debye length goes to zero.To establish this limit mathematically rigorously,we derive uniform(in Debye length)energy estimates for the remainders.Due to the quantum effects in the mass equation and the energy equation,we need to deal with the higher order derivatives and construct suitable energy norms.By applying a formal expansion according to Debye length and a deep analysis of the structure of the full quantum Navier-Stokes-Poisson equations,we study the asymptotic behaviors mathematically rigorously.In Chapter 3,we disscuss the quasineutral limit for the compressible quantum Navier-Stokes-Maxwell equations with heat conductivity in a torus 3?.The model is a complex model coupled through the Navier-Stokes equation and the Maxwell equation via the Lorentz force.In the first part,we study rigorously the asymptotic behaviors of this model for the well-prepared initial data.Based on the special structure of the equations combined with curl-div decomposition of the gradient and the wave-type equation of the Maxwell equation,we establish rigorous uniform estimates on the error functions with respect to the Debye length.It is proved rigorously that as the Debye length tends to zero,the solutions of the quantum compressible Navier-Stokes-Maxwell equations converge to the solutions of the incompressible e-MHD equations.For the ill-prepared initial data,we use the multiple-scale expansion formalism,singular perturbation methods and sublinear growth condition to prove the convergence of strong solutions for the quantum compressible Navier-Stokes-Maxwell equations towards those for the incompressible e-MHD equations plus the fast singular oscillating velocity field and electric field as the Debye length goes to zero.In Chapter 4,we study the quasineutral limit for the isentropic compressible bipolar Euler-Maxwell equations for well-prepared initial data.Based on the asymptotic behaviors,curl-div decomposition of the gradient,the iteration techniques and the standard compactness argument,we close the estimate for the error system in the proof.Furthermore,we prove rigorously the solutions to the isentropic bipolar Euler-Maxwell equations converge globally in time up to the maximal existence time of the smooth solutions to the compressible Euler equation.In Chapter 5,we study the long wavelength limit for the hydromagnetic waves in cold plasma.Based on the Gardner-Morikawa transform and the reductive perturbation theory,it is demonstrated that on very long time intervals,the solutions of such hydromagnetic waves converge to the solution of the Korteweg-de Vries equation(Kd V)as the scale goes to zero.In Chapter 6,we consider the existence of smooth solutions to the three dimensional non-isentropic compressible quantum hydrodynamic equations without visocity.In the first part,we consider the existence and uniqueness of a time periodic solution to this model.By using a regularized approximation scheme,Leray-Schauder theory together with the uniform estimates(in the domain and the man-added positive constant),we derive the existence of time periodic solutions in a bounded domain with periodic boundary under some smallness and structure assumption imposed on the external force.Then based on a limiting method and a diagonal argument,we extend the result to the unbounded domain.In the second part,we deal with the existence of global smooth solutions of the three dimensional non-isentropic quantum hydrodynamic equations without visocity over a bounded domain.The boundary conditions finally adopted are the insulating boundary conditions.We establish the uniform a priori estimates provided that the initial perturbation around a constant state is small enough together with the density and the temperature are positive.Finally,combined the local exiatence theory with continuation argument,we prove the global solutions to the initial boundary value problem of this model in a bounded domain.In Chapter 7,we summarize the main results in our thesis and propose some problems in the future.