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The Problem of Sonic Shock Formation
Ying, Hao

2015, Doctor of Philosophy, Ohio State University, Mathematics.
The equations of compressible ideal gas flow in Eulerian coordinates and simplified models of this flow are studied by many mathematicians, physicists and engineers, because of their rich mathematical structure and their close relation to many physics and engineering problems. The Eulerian system of quasilinear PDEs has the feature that smooth solutions do not exist to most problems; one must study weak solutions, which typically contain shocks. While shocks are quite well understood in one space dimension, little is known about the behavior of shocks in multi-dimensional space.

Since Canic, Keyfitz and Lieberman's seminal paper [10], there has been much research on multi-dimensional shocks using self-similar reduction, the free boundary approach and the procedure introduced in [10]. A feature of these reduced systems is that the systems change type from hyperbolic to elliptic in some regions of space. Since quasilinear elliptic equations have smooth solutions, the shock cannot appear inside of the elliptic region. A natural question to ask is whether the shock can form at the sonic line (the curve separating regions in which the system is of different types).

In the first part of the dissertation, we try to give a positive answer to this question. We use the unsteady transonic small disturbance equation (UTSDE) as a model system and choose a configuration where it appears that the shock forms at the sonic line. We follow the procedure of [10] to prove the existence of a solution to this problem. To implement this procedure, we face some new technical challenges. First, the second order elliptic PDE developed from the UTSDE is degenerate on the sonic line. Second, the oblique derivative condition posed on the shock becomes tangential at the shock formation point. Finally, the first two difficulties happen at the same time at the shock formation point. To overcome those difficulties, we first introduce several cut-off functions and small parameters to regularize the problem. Then we construct several barrier functions to remove the cut-off functions after solving the regularized problem. Finally, we use the Arzela-Ascoli Theorem and a diagonalization argument to obtain the solution to the original problem by sending the small parameters to zero.

In the second part of the dissertation, we study a situation that arises in shock diffraction by a 90 degree corner, using the isentropic Euler equations. In preparing to extend the method in the first part to this problem, we need to resolve a new difficulty that the PDEs in this system become degenerate at “stagnation points”. To gain some understanding of the behavior of the solutions near stagnation points we study the behavior of the solutions to the isentropic Euler equations near stagnation points under the self-similar and axisymmetric assumptions. We find a solution to the self-similar and axisymmetric Euler equations with a stagnation point at the origin with certain asymptotic behavior which gives some insights of how solutions behave near stagnation points.
Barbara Keyfitz (Advisor)
Fei-Ran Tian (Committee Member)
Chuan Xue (Committee Member)
Yuri Kovchegov (Other)
88 p.

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