| If the flow velocity is given by ∇xϕ for a function ϕ, then such a flow is called a potential flow . This thesis contains two transonic shock problems of the potential flow: (i) regular shock reflection by a wedge and (ii) transonic shocks in a divergent nozzle.;When a normal shock hits a symmetric wedge of a large angle, a regular shock reflection occurs. In [CF4, CF5], the existence of a global solution to the shock reflection is proven for the isentropic potential flow and the velocity potential phi has an expression of phi(x, t) = tpsi( xt ) for t > 0. The first part of the thesis proves that psi( xt ) is at most C1,1 with respect to xt in the region bounded by the reflected shock and the boundary of the wedge. For that, this thesis proves C2,alpha regularity of solutions to a class of nonlinear degenerate elliptic equations up to an ellipticity degenerate boundary for alpha ∈ (0, 1).;The second part introduces the non-isentropic potential flow model and proves the unique existence of a transonic shock in a multidimensional divergent nozzle for a fixed pressure at the exit. The result contains the unique solvability of a class of transport equations with velocity vector fields of a weak regularity, an inverse mapping theorem in infinite dimensional Banach spaces without continuous differentiability of a mapping and the unique solvability of elliptic equations with a Neumann boundary condition blowing up at corners of the nozzle. |
