Existence And Asymptotic Behavior Of Solutions Of Several Kinds Of Hydrodynamic Correlation Equations

In t.his dissertation,micropolar fluid equations,Keller-Segel-Stokes equations and a class of non-isentropic hydrodynamic equations with quantum effects for semiconductors are considered.As a class of important nonlinear partial differential equations,the microp-olar fluid equations describes a class of non-Newtonian fluid motions with micro-rotational effects and inertial force involved,which can well represent the dy namic behavior of some incompressible fluid that cannot be described by the classical Navier-Stokes model,such as the movement of animal bloodliquid crystal and the flow of dilute water-soluble poly-mer solution.In nature,organisms are everywhere,and their dynamic behaviors tend to show certain chemotactic phenomena to some natural factors.Organisms in the fluid will inevitably be affected by the fluid motion,and the Keller-Segel-Stokes equations can well describe the chemotactic movement of organisms in Stokes fluids.The quantum hydro-dynamic equations play an important and irreplaceable role in modeling the motion of electrons or hole transports under self-consistent electric fields.The major advantage of macroscopic quantum hydrodynamic models lies in that they are able to describe directly the dynamic evolution of physical observable variables and hence largely facilitate simu-lations of quantum phenomena.The quantum hydrodynamic equations are useful in the modelling of semiconductor devices in nano-size,such as High Electron Mobility Transis-tor(HEMT),Metal-Oxide-Semiconductor Field-Effect Transistor(MOSFET),Resonant Tunneling Diode(RTD).This dissertation will study the above three types of equations,arranged as following:In Chapter 1,we mainly introduce the relevant background,current research status,research objectives and results of the micropolar fluid equations,the Keller-Segel-Stokes equations and the quantum hydrodynamic equations for semiconductors,and make some necessary preliminariesIn Chapter 2,we investigate the global well-posedness and asymptotic behavior of solutions of the non-autonomous micropolar fluid equations in 2D domains.(?)In 2D bounded domains:First,,the existence of a compact pullback absorbing family is proved by combining the energy and semigroup method wit,h ?-regularity theory and the compact embedding between spaces.Then,the existence and regularity of pullback attractor in H and V are obtained by proving the flattening property of the generated evolution process using energy method.(?)In 2D unbounded domains:First,applying the technique of truncatio n functions,decomposition of spatial domain,and energy method,we show the existence of the pull-back attractor,and further verify the tempered behavior and uper semicontinuity of the pullback attractor.Then,utilizing the Lax-Milgram theorem,Brouwer fixed point theorem and limit thought,we obtain the existence and uniqueness of steady-state solution of the micropolar fluid equations with delay,and further investigate the stability of the steady-state solution through the energy method.Finally,we make use of truncation functions,decomposition of spatial domain,and the Galerkin method to prove the well-posedness of the global solution for the micropolar fluid equations with delay.In Chapter 3,we study the asymptotic behavior of the Keller-Segel-Stokes equations with arbitrary porous medium diffusion in 2D bounded domains.Invoking energy method and the embedding between spaces,we obtain the existence of trajectory attractor and generalized global attractor.In Chapter 4,we investigate the existence and asymptotic behavior of solutions of the non-isentropic viscous quantum hydrodynamic equations for semiconductors in one dimensional real line.First,we prove the existence and uniqueness of steady-state solution of the equations by the energy method and continuity method.Then,by using iterative method,we show the existence and uniqueness of the classical solution and the stability of the steady-state solution.