On the regularity of the velocity potential in two-dimensional transonic flow
type: Item
instance of: scholarly article
title: On the regularity of the velocity potential in two-dimensional transonic flow
zbMATH Open document ID: 1007.35065
author: Jong Uhn Kim
published in: Differential and Integral Equations
publication date: 2000-05-24T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35M10
Mathematics Subject Classification ID: 76H05
zbMATH DE Number: 1452487
zbMATH Keywords: transonic flow dynamics
zbMATH Keywords: sonic line
zbMATH Keywords: regularity of tangential and normal derivatives
zbMATH Keywords: degenerate hyperbolic equation
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1452487