# Continuous Subsonic-Sonic Flows in a Convergent Nozzle

- 41 Downloads

## Abstract

This paper concerns continuous subsonic-sonic potential flows in a two-dimensional convergent nozzle. It is shown that for a given nozzle which is a perturbation of a straight one, a given point on its wall where the curvature is zero, and a given inlet which is a perturbation of an arc centered at the vertex, there exists uniquely a continuous subsonic-sonic flow whose velocity vector is along the normal direction at the inlet and the sonic curve, which satisfies the slip conditions on the nozzle walls and whose sonic curve intersects the upper wall at the given point. Furthermore, the sonic curve of this flow is a free boundary, where the flow is singular in the sense that the speed is only *C*^{1/2} Hölder continuous and the acceleration blows up. The perturbation problem is solved in the potential plane, where the flow is governed by a free boundary problem of a degenerate elliptic equation with two free boundaries and two nonlocal boundary conditions, and the equation is degenerate at one free boundary.

## Keywords

Continuous subsonic-sonic flow free boundary nonlocal boundary condition degeneracy singularity## MR(2010) Subject Classification

35R35 76N10 35J70## Preview

Unable to display preview. Download preview PDF.

## References

- [1]Bers, L.: Existence and uniqueness of a subsonic flow past a given profile.
*Comm. Pure Appl. Math.*,**7**(3), 441–504 (1954)MathSciNetCrossRefzbMATHGoogle Scholar - [2]Bers, L.:
*Mathematical Aspects of Subsonic and Transonic Gas Dynamics*, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958zbMATHGoogle Scholar - [3]Chen, G. Q., Dafermos, C. M., Slemrod, M., et al.: On two-dimensional sonic-subsonic flow.
*Comm. Math. Phys.*,**271**(3), 635–647 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - [4]Courant, R., Friedrichs, K. O.:
*Supersonic Flow and Shock Waves*, Interscience Publishers, Inc., New York, NY, 1948zbMATHGoogle Scholar - [5]Du, L. L., Xie, C. J., Xin, Z. P.: Steady subsonic ideal flows through an infinitely long nozzle with large vorticity.
*Commun. Math. Phys.*,**328**(1), 327–354 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - [6]Nie, Y. Y., Wang, C. P.: Continuous subsonic-sonic flows in convergent nozzles with straight solid walls.
*Nonlinearity*,**29**(1), 86–130 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - [7]Oleienik, O. A., Radkevic, E. V.:
*Second Order Differential Equations with Nonnegative Characteristic Form, American Mathematical Society*, Rhode Island and Plenum Press, New York, 1973CrossRefGoogle Scholar - [8]Wang, C. P.: Continuous subsonic-sonic flows in a general nozzle.
*J. Differential Equations*,**259**(7), 2546–2575 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - [9]Wang, C. P., Xin, Z. P.: Optimal Hölder continuity for a class of degenerate elliptic problems with an application to subsonic-subsonic flows.
*Comm. Partial Differential Equations*,**36**(5), 873–924 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - [10]Wang, C. P., Xin, Z. P.: On a degenerate free boundary problem and continuous subsonic-sonic flows in a convergent nozzle.
*Arch. Ration. Mech. Anal.*,**208**(3), 911–975 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - [11]Wang, C. P., Xin, Z. P.:
*Smooth transonic flows in de Laval nozzles*. arXiv preprint, arXiv:1304.2473, 2013Google Scholar - [12]Wang, C. P., Xin, Z. P.: Global smooth supersonic flows in infinite expanding nozzles.
*SIAM J. Math. Anal.*,**47**(4), 3151–3211 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - [13]Wang, C. P., Xin, Z. P.: On sonic curves of smooth subsonic-sonic and transonic flows.
*SIAM J. Math. Anal.*,**48**(4), 2414–2453 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - [14]Xie, C. J., Xin, Z. P.: Global subsonic and subsonic-sonic flows through infinitely long nozzles.
*Indiana Univ. Math. J.*,**56**(6), 2991–3023 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - [15]Xie, C. J., Xin, Z. P.: Existence of global steady subsonic Euler flows through infinitely long nozzles.
*SIAM J. Math. Anal.*,**42**(2), 751–784 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - [16]Xie, C. J., Xin, Z. P.: Global subsonic and subsonic-sonic flows through infinitely long axially symmetric nozzles.
*J. Differential Equations*,**248**(11), 2657–2683 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - [17]Yin, J. X., Wang, C. P.: Evolutionary weighted
*p*-Laplacian with boundary degeneracy.*J. Differential Equations*,**237**(2), 421–445 (2007)MathSciNetCrossRefzbMATHGoogle Scholar