On the regularity of the velocity potential in two-dimensional transonic flow

Abstract

We present a new result on the regularity of the velocity potential in transonic gas dynamics of two space dimensions. We prove that the velocity potential is $C^{\infty}$ in a neighborhood of a sonic point if the sonic line is locally $C^{\infty}, $ and if the velocity potential is known to be locally $C^4. $ Our main tool is Hörmander's energy method for the Tricomi operator, the commutator estimate of Kato and Ponce, and a result of Berezin for a degenerate hyperbolic equation.