Initial-boundary value problems in fluid dynamics

This thesis is devoted to studies of initial-boundary value problems (IBVPs) for systems of partial differential equations (PDEs) arising from fluid mechanics modeling, especially for the compressible Euler equations with frictional damping, the Boussinesq equations, the Cahn-Hilliard equations and the incompressible density-dependent Navier-Stokes equations. The emphasis of this thesis is to understand the influences to the qualitative behavior of solutions caused by boundary effects and various dissipative mechanisms including damping, viscosity and heat diffusion. We will present results concerning global existence and large-time asymptotic behavior of solutions to miscellaneous initial-boundary value problems. The results obtained consist of three parts.The Part 1, containing Chapters II--III, is concerned with the study of compressible Euler equations with frictional damping. In Chapter II, we first construct global Linfinity entropy weak solutions to the IBVP for one-dimensional damped compressible Euler equations on bounded domains with physical boundaries. Time asymptotically, the density is conjectured to satisfy the porous medium equation (PME) and the momentum obeys to the classical Darcy's law. Based on entropy principle, we show that the physical solution converges to steady states exponentially fast in time. We also prove that the same is true for the related IBVP of porous medium equation provided that the two systems carry the same initial mass and thus justify the validity of Darcy's law in large time. In Chapter III, we continue the study of damped compressible Euler equations on bounded domains. We prove global existence and uniqueness of classical solutions to the IBVP for three-dimensional damped compressible Euler equations on bounded domains with the slip boundary condition when the initial data is near its equilibrium. Furthermore, based on energy estimate, we show that the classical solution is captured by that of the porous medium equation exponentially fast as time tends to infinity and justify Darcy's law in large time.In Part 2, we study the two-dimensional Boussinesq equations with partial viscosity. In Chapter IV, we first prove global existence of smooth solutions to the IBVP for the viscous non-heat-conductive Boussinesq equations on bounded domains with arbitrary smooth initial data and the no-slip boundary condition. In addition, the uniform bound of the kinetic energy is obtained as a by product. Then we study the IBVP for another type of 2D Boussinesq equations with partial viscosity which is inviscid and heat-conductive. We show that there exists a unique global smooth solution to the IBVP for arbitrary smooth initial data and for physical boundary conditions. Furthermore, due to dissipation and boundary effects, we prove that the kinetic energy is uniformly bounded in time and the temperature converges exponentially to a constant state which is the value of the temperature on the boundary of the domain. The results obtained in this part suggest that the partial dissipative mechanism is indeed strong enough to compensate the effects of gravitational force and nonlinear convection in order to prevent the development of singularity in the systems.Part 3 is contributed to the mathematical analysis of multi-phase/mixing flows. In Chapter V, we first study the IBVP for a system of PDEs obtained by coupling the Cahn-Hilliard equation and the two-dimensional Boussinesq equations which stands for a model of a multi-phase flow under shear and the influence of gravitational force. Then we study a model of a two-component mixture, with a diffusive mass exchange among the medium particles of various density accounted for, which is closely related to the 2D incompressible density-dependent Navier-Stokes equations. For both systems of equations, we prove global existence of smooth solutions to the IBVPs with arbitrary smooth initial data and physical boundary conditions.