On Multi-dimensional Steady Subsonic Flows Determined by Physical Boundary Conditions

In this thesis, we investigate an inflow-outflow problem for subsonic gas flows in a nozzle with finite length, aiming at finding intrinsic (physically acceptable) boundary conditions on upstream and downstream. We first characterize a set of physical boundary conditions that ensure the existence and uniqueness of a subsonic irrotational flow in a rectangle. Our results show that suppose we prescribe the horizontal incoming flow angle at the inlet and an appropriate pressure at the exit, there exists two positive constants m 0 and m1 with m0 < m1, such that a global subsonic irrotational flow exists uniquely in the nozzle, provided that the incoming mass flux m ∈ [m0, m 1). The maximum speed will approach the sonic speed as the mass flux m tends to m1. The new difficulties arise from the nonlocal term involved in the mass flux and the pressure condition at the exit. We first introduce an auxiliary problem with the Bernoulli's constant as a parameter to localize the nonlocal term and then establish a monotonic relation between the mass flux and the Bernoulli's constant to recover the original problem. To deal with the loss of obliqueness induced by the pressure condition at the exit, we employ the formulation in terms of the angular velocity and the density. A Moser iteration is applied to obtain the Linfinity estimate of the angular velocity, which guarantees that the flow possesses a positive horizontal velocity in the whole nozzle.;As a continuation, we investigate the influence of the incoming flow angle and the geometry structure of the nozzle walls on subsonic flows in a finitely long curved nozzle. It turns out to be interesting that the incoming flow angle and the angles of inclination of nozzle walls play the same role as the end pressure. The curvatures of the nozzle walls play an important role. We also extend our results to subsonic Euler flows in the 2-D and 3-D asymmetric cases.;Then it comes to the most interesting and difficult case--the 3-D subsonic Euler flow in a bounded nozzle, which is also the essential part of this thesis. The boundary conditions we have imposed in the 2-D case have a natural extension in the 3-D case. These important clues help us a lot to develop a new formulation to get some insights on the coupling structure between hyperbolic and elliptic modes in the Euler equations. The key idea in our new formulation is to use the Bernoulli's law to reduce the dimension of the velocity field by defining new variables &parl0;1,b2=u2u 1,b3=u3 u1&parr0; and replacing u1 by the Bernoulli's function B through u21=2B-h r1+ b22+b23 . In this way, we can explore the role of the Bernoulli's law in greater depth and hope that may simplify the Euler equations a little bit. We find a new conserved quantity for flows with a constant Bernoulli's function, which behaves like the scaled vorticity in the 2-D case. More surprisingly, a system of new conservation laws can be derived, which is never been observed before, even in the two dimensional case. We employ this formulation to construct a smooth subsonic Euler flow in a rectangular cylinder by assigning the incoming flow angles and the Bernoulli's function at the inlet and the end pressure at the exit, which is also required to be adjacent to some special subsonic states. The same idea can be applied to obtain similar information for the incompressible Euler equations, the self-similar Euler equations, the steady Euler equations with damping, the steady Euler-Poisson equations and the steady Euler-Maxwell equations.;Last, we are concerned with the structural stability of some steady subsonic solutions for the Euler-Poisson system. A steady subsonic solution with subsonic background charge is proven to be structurally stable with respect to small perturbations of the background charge, the incoming flow angles and the end pressure, provided the background solution has a low Mach number and a small electric field. The new ingredient in our mathematical analysis is the solvability of a new second order elliptic system supplemented with oblique derivative conditions at the inlet and Dirichlet boundary conditions at the exit of the nozzle.