For more information about this meeting, contact Becky Halpenny.

Title: | "The structure of solutions near a degenerate line for mixed type equations" |

Seminar: | Ph.D. Thesis Defense |

Speaker: | Tianyou Zhang, Adviser: Yuxi Zheng |

Abstract: |

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. |

### Room Reservation Information

Room Number: | MB106 |

Date: | 04 / 30 / 2012 |

Time: | 10:30am - 12:30pm |