Math 2.0

Fluid mechanics Compressible flow and gas dynamics Theorem

Morawetz theorem on shockless airfoils

Builds on Transonic flowPotential flowIsentropic flowHodograph transformationTricomi equationShock wave

Statement

Consider steady, plane, inviscid, irrotational and isentropic flow of a compressible gas past a profile PP, with a subsonic uniform stream at infinity. Suppose that past PP there is a continuous transonic flow, that is, a flow without shocks. Suppose its supersonic region is bounded and adjacent to the profile. Then this flow is not stable under changes of the profile. There are arbitrarily small smooth changes of PP, made only on the arc of PP that bounds the supersonic region, with the following property. The changed profile admits no continuous transonic flow close to the original one.

Remarks

The idea of the proof is to pass to the hodograph plane by the hodograph transformation. There the potential flow equation becomes linear and of mixed type, like the Tricomi equation. The perturbation of the flow then solves a linear boundary value problem with data on the whole boundary of the region, including its hyperbolic part. Morawetz showed that this problem is overdetermined, so that for general perturbations of the profile it has no solution. So a shockless transonic profile is an exceptional, isolated design, and generic small changes create shocks. Shockless airfoils can still be designed, but each one works only at its design conditions.

Going deeper

For a student

The result explains why steady transonic flows past airfoils must in general be sought as weak solutions with shocks. One solves the full potential or Euler equations in this weak sense, rather than looking for smooth solutions. It is a statement about the ill-posedness of a boundary value problem for a mixed-type equation, and it motivated much of the later theory of such problems. It also guides numerical methods for transonic flow, which must be able to capture shocks rather than assume smoothness.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph
Hodograph transformation, concept, no evidence levelHodograph transformat…conceptIsentropic flow, concept, no evidence levelIsentropic flowconceptPotential flow, concept, no evidence levelPotential flowconceptShock wave, concept, no evidence levelShock waveconceptTransonic flow, concept, no evidence levelTransonic flowconceptTricomi equation, concept, no evidence levelTricomi equationconceptMorawetz theorem on shockless airfoils, statement, no evidence levelMorawetz theorem on s…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Hodograph transformation concept
  2. Isentropic flow concept
  3. Potential flow concept
  4. Shock wave concept
  5. Transonic flow concept
  6. Tricomi equation concept
  7. Morawetz theorem on shockless airfoils statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Cathleen_Synge_Morawetz
About this page
Tag4MWYZQC697
URIm2://4MWYZQC697
Kindstatement, theorem
WritingW0 stub
MSC76H05, 35M10
StatusThis copy of the site has no build data. Run m2 status 4MWYZQC697.
AboutTransonic flow
UsesTransonic flow, Potential flow, Isentropic flow, Hodograph transformation, Tricomi equation, Shock wave
Changed3 November 2026

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