Free boundary problems for the unsteady transonic small disturbance equation: Transonic regular reflection

Weak Shock Reflection Modeled by the Unsteady Transonic Small Disturbance Equation

Rarefaction wave interaction for the unsteady transonic small disturbance equations

We study a Riemann problem for the unsteady transonic small disturbance equations that results in a diverging rarefaction problem. The self-similar reduction leads to a boundary value problem with equations that change type (hyperbolic-elliptic) and a sonic line that is a free boundary. We summarize the principal ideas and present the main features of the problem. The flow in the hyperbolic par...

An Elliptic Problem Arising from the Unsteady Transonic Small Disturbance Equation

We prove a theorem on existence of a weak solution of the Dirich-let problem for a quasilinear elliptic equation with a degeneracy on one part of the boundary. The degeneracy is of a type (\Keldysh type") associated with singular behavior | blow-up of a derivative | at the boundary. We deene an associated operator which is continuous , pseudo-monotone and coercive and show that a weak solution ...

Free boundary problems in shock reflection/diffraction and related transonic flow problems.

Shock waves are steep wavefronts that are fundamental in nature, especially in high-speed fluid flows. When a shock hits an obstacle, or a flying body meets a shock, shock reflection/diffraction phenomena occur. In this paper, we show how several long-standing shock reflection/diffraction problems can be formulated as free boundary problems, discuss some recent progress in developing mathematic...

Constraints on Possible Singularities for the Unsteady Transonic Small Disturbance (UTSD) Equations

We discuss the singular behavior of solutions to two-dimensional, general secondorder, uniformly elliptic equations in divergence form, with bounded measurable coefficients and discontinuous Dirichlet data along a portion of a Lipschitz boundary. We show that the conjugate to the solution develops a singularity that is at least logarithmic along the boundary at the points of discontinuity in th...