On the regularity of the velocity potential in two-dimensional transonic flow (Q1979109)

The considered equation \((c^2 - \Phi ^2_x)\Phi _{xx} - 2\Phi _x \Phi _y \Phi _{xy} + (c^2 - \Phi ^2_y)\Phi _{yy} = 0\) with \(c^2 = 1-(\gamma -1)|\nabla \Phi |^2/2\) \((\gamma >1)\) is elliptic if \(|\nabla \Phi |^2 < c^2\) and hyperbolic if \(|\nabla \Phi |^2 >c^2\). This type of equation occurs in the transonic flow dynamics, \(\Phi \) is then the velocity potential. The main result gives the following information about a property of the sonic line \(\Gamma \) consisting of points where \(|\nabla \Phi |^2=c^2\): Provided that \(\Phi \) is \(C^4\) in a neighborhood of \((x_0, y_0) \in \Gamma \) and this point is not `exceptional' then \(\Phi \) is \(C^{\infty }\) there. This is done by studying regularity of tangential and normal derivatives in subsonic and supersonic regions near the sonic line which is combined with the definition of the sonic line. (Here the restriction to two dimensions seems crucial.) The commutator estimates of Kayo and Ponce as well as a Berezin result for degenerate hyperbolic equation are used in the proof.