# Meeting Details

Title: "The structure of solutions near a degenerate line for mixed type equations" Ph.D. Thesis Defense Tianyou Zhang, Adviser: Yuxi Zheng In two dimensional physical space, the Euler equations and its simplified model Pressure Gradient (PG) equations are mixed type equations, which change type from hyperbolic to elliptic across the sonic curve on the self-similar plane. The regularity of solutions near the sonic curve is an important issue. We consider a new type of problems. Given a smooth curve as the sonic curve for the PG system in the self-similar plane, we assign suitable boundary conditions for the pressure $p$ on it and build a class of regular solutions extending from the sonic line to the hyperbolic region. The key in the construction is a novel change of coordinates, using both the state and the space-time variables. With it, we mange to separate the singular terms in the analysis. The idea obtained on the PG equations stimulates us to consider the 2-D Steady Euler equations, which are type-changing and model the steady mixed flow. This time, we fix a smooth sonic curve, assign boundary conditions for the sound speed $c$ on it and obtain the existence of classical supersonic solutions on one side of the curve. We use a new set of coordinates, which are not explored in literature, to deal with the degeneracy of the system at the sonic curve.