American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Kuramoto order parameters and phase concentration for the Kuramoto-Sakaguchi equation with frustration
  • CPAA Home
  • This Issue
  • Next Article
    Uniqueness of steady 1-D shock solutions in a finite nozzle via vanishing viscosity aguments
July & August  2021, 20(7&8): 2555-2577. doi: 10.3934/cpaa.2021070

A degenerate elliptic problem from subsonic-sonic flows in convergent nozzles

Guanming Gai , Yuanyuan Nie , and  Chunpeng Wang

School of Mathematics, Jilin University, Changchun 130012, Jilin, China

* Corresponding author

Received  December 2020 Revised  March 2021 Published  July & August 2021 Early access  April 2021

Fund Project: Supported by grants from the National Natural Science Foundation of China (Nos. 11871133 and 11925105)

This paper concerns continuous subsonic-sonic potential flows in a two dimensional convergent nozzle, which is governed by a free boundary problem of a quasilinear degenerate elliptic equation. It is shown that for a given nozzle which is a perturbation of an straight one, and a given mass flux, there exists uniquely a continuous subsonic-sonic flow whose velocity vector is along the normal direction at the inlet and the sonic curve. 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 at the sonic state.

Keywords: Subsonic-sonic flow, free boundary, degeneracy.
Mathematics Subject Classification: 35R35, 76N10, 35J70.
Citation: Guanming Gai, Yuanyuan Nie, Chunpeng Wang. A degenerate elliptic problem from subsonic-sonic flows in convergent nozzles. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2555-2577. doi: 10.3934/cpaa.2021070
References:
[1]

L. Bers, Existence and uniqueness of a subsonic flow past a given profile, Commun. Pure Appl. Math., 7 (1954), 441-504.  doi: 10.1002/cpa.3160070303.

[2]

L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, John Wiley & Sons, Inc., New York, Chapman & Hall, Ltd., London, 1958.

[3]

C. Chen, L. L. Du, C. J. Xie and Z. P. Xin, Two dimensional subsonic Euler flows past a wall or a symmetric body, Arch. Ration. Mech. Anal., 221 (2016), 559-602.  doi: 10.1007/s00205-016-0968-0.

[4]

G. Q. Chen, C. M. Dafermos, M. Slemrod and D. H. Wang, On two-dimensional sonic-subsonic flow, Commun. Math. Phys., 271 (2007), 635-647.  doi: 10.1007/s00220-007-0211-9.

[5]

G. Q. Chen, F. M. Huang and T. Y. Wang, Subsonic-sonic limit of approximate solutions to multidimensional steady Euler equations, Arch. Ration. Mech. Anal., 219 (2016), 719-740.  doi: 10.1007/s00205-015-0905-7.

[6]

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, NY, 1948.

[7]

G. C. Dong and B. Ou, Subsonic flows around a body in space, Commun. Partial Differ. Equ., 18 (1993), 355-379.  doi: 10.1080/03605309308820933.

[8]

L. L. Du, C. J. Xie and Z. P. Xin, Steady subsonic ideal flows through an infinitely long nozzle with large vorticity, Commun. Math. Phys., 328 (2014), 327-354.  doi: 10.1007/s00220-014-1951-y.

[9]

L. L. Du, Z. P. Xin and W. Yan, Subsonic flows in a multi-dimensional nozzle, Arch. Ration. Mech. Anal., 201 (2011), 965-1012.  doi: 10.1007/s00205-011-0406-2.

[10]

R. Finn and D. Gilbarg, Three-dimensional subsonic flows, and asymptotic estimates for elliptic partial differential equations, Acta Math., 98 (1957), 265-296.  doi: 10.1007/BF02404476.

[11]

G. M. Lieberman, Local estimates for subsolutions and supersolutions of oblique derivative problems for general second order elliptic equations, Trans. Amer. Math. Soc., 304 (1987), 343-353.  doi: 10.1090/S0002-9947-1987-0906819-0.

[12]

Y. Y. Nie and C. P. Wang, Continuous subsonic-sonic flows in convergent nozzles with straight solid walls, Nonlinearity, 29 (2016), 86-130.  doi: 10.1088/0951-7715/29/1/86.

[13]

Y. Y. Nie and C. P. Wang, Continuous subsonic-sonic flows in a convergent nozzle, Acta Math. Sin., 34 (2018), 749–772. doi: 10.1007/s10114-017-7341-6.

[14]

C. P. Wang, Continuous subsonic-sonic flows in a general nozzle, J. Differ. Equ., 259 (2015), 2546-2575.  doi: 10.1016/j.jde.2015.03.036.

[15]

C. P. Wang and Z. P. Xin, On a degenerate free boundary problem and continuous subsonic-sonic flows in a convergent nozzle, Arch. Ration. Mech. Anal., 208 (2013), 911-975.  doi: 10.1007/s00205-012-0607-3.

[16]

C. P. Wang and Z. P. Xin, Global smooth supersonic flows in infinite expanding nozzles, SIAM J. Math. Anal., 47 (2015), 3151-3211.  doi: 10.1137/140994289.

[17]

C. P. Wang and Z. P. Xin, On sonic curves of smooth subsonic-sonic and transonic flows, SIAM J. Math. Anal., 48 (2016), 2414-2453.  doi: 10.1137/16M1056407.

[18]

C. P. Wang and Z. P. Xin, Smooth transonic flows of Meyer type in de Laval nozzles, Arch. Ration. Mech. Anal., 232 (2019), 1597-1647.  doi: 10.1007/s00205-018-01350-9.

[19]

C. P. Wang and M. J. Zhou, A degenerate elliptic problem from subsonic-sonic flows in general nozzles, J. Differ. Equ., 267 (2019), 3778-3796.  doi: 10.1016/j.jde.2019.04.026.

[20]

C. J. Xie and Z. P. Xin, Global subsonic and subsonic-sonic flows through infinitely long nozzles, Indiana U. Math. J., 56 (2007), 2991-3023.  doi: 10.1512/iumj.2007.56.3108.

[21]

C. J. Xie and Z. P. Xin, Existence of global steady subsonic Euler flows through infinitely long nozzles, SIAM J. Math. Anal., 42 (2010), 751-784.  doi: 10.1137/09076667X.

[22]

J. X. Yin and C. P. Wang, Evolutionary weighted $p$-Laplacian with boundary degeneracy, J. Differ. Equ., 237 (2007), 421-445.  doi: 10.1016/j.jde.2007.03.012.

show all references

References:
[1]

L. Bers, Existence and uniqueness of a subsonic flow past a given profile, Commun. Pure Appl. Math., 7 (1954), 441-504.  doi: 10.1002/cpa.3160070303.

[2]

L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, John Wiley & Sons, Inc., New York, Chapman & Hall, Ltd., London, 1958.

[3]

C. Chen, L. L. Du, C. J. Xie and Z. P. Xin, Two dimensional subsonic Euler flows past a wall or a symmetric body, Arch. Ration. Mech. Anal., 221 (2016), 559-602.  doi: 10.1007/s00205-016-0968-0.

[4]

G. Q. Chen, C. M. Dafermos, M. Slemrod and D. H. Wang, On two-dimensional sonic-subsonic flow, Commun. Math. Phys., 271 (2007), 635-647.  doi: 10.1007/s00220-007-0211-9.

[5]

G. Q. Chen, F. M. Huang and T. Y. Wang, Subsonic-sonic limit of approximate solutions to multidimensional steady Euler equations, Arch. Ration. Mech. Anal., 219 (2016), 719-740.  doi: 10.1007/s00205-015-0905-7.

[6]

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, NY, 1948.

[7]

G. C. Dong and B. Ou, Subsonic flows around a body in space, Commun. Partial Differ. Equ., 18 (1993), 355-379.  doi: 10.1080/03605309308820933.

[8]

L. L. Du, C. J. Xie and Z. P. Xin, Steady subsonic ideal flows through an infinitely long nozzle with large vorticity, Commun. Math. Phys., 328 (2014), 327-354.  doi: 10.1007/s00220-014-1951-y.

[9]

L. L. Du, Z. P. Xin and W. Yan, Subsonic flows in a multi-dimensional nozzle, Arch. Ration. Mech. Anal., 201 (2011), 965-1012.  doi: 10.1007/s00205-011-0406-2.

[10]

R. Finn and D. Gilbarg, Three-dimensional subsonic flows, and asymptotic estimates for elliptic partial differential equations, Acta Math., 98 (1957), 265-296.  doi: 10.1007/BF02404476.

[11]

G. M. Lieberman, Local estimates for subsolutions and supersolutions of oblique derivative problems for general second order elliptic equations, Trans. Amer. Math. Soc., 304 (1987), 343-353.  doi: 10.1090/S0002-9947-1987-0906819-0.

[12]

Y. Y. Nie and C. P. Wang, Continuous subsonic-sonic flows in convergent nozzles with straight solid walls, Nonlinearity, 29 (2016), 86-130.  doi: 10.1088/0951-7715/29/1/86.

[13]

Y. Y. Nie and C. P. Wang, Continuous subsonic-sonic flows in a convergent nozzle, Acta Math. Sin., 34 (2018), 749–772. doi: 10.1007/s10114-017-7341-6.

[14]

C. P. Wang, Continuous subsonic-sonic flows in a general nozzle, J. Differ. Equ., 259 (2015), 2546-2575.  doi: 10.1016/j.jde.2015.03.036.

[15]

C. P. Wang and Z. P. Xin, On a degenerate free boundary problem and continuous subsonic-sonic flows in a convergent nozzle, Arch. Ration. Mech. Anal., 208 (2013), 911-975.  doi: 10.1007/s00205-012-0607-3.

[16]

C. P. Wang and Z. P. Xin, Global smooth supersonic flows in infinite expanding nozzles, SIAM J. Math. Anal., 47 (2015), 3151-3211.  doi: 10.1137/140994289.

[17]

C. P. Wang and Z. P. Xin, On sonic curves of smooth subsonic-sonic and transonic flows, SIAM J. Math. Anal., 48 (2016), 2414-2453.  doi: 10.1137/16M1056407.

[18]

C. P. Wang and Z. P. Xin, Smooth transonic flows of Meyer type in de Laval nozzles, Arch. Ration. Mech. Anal., 232 (2019), 1597-1647.  doi: 10.1007/s00205-018-01350-9.

[19]

C. P. Wang and M. J. Zhou, A degenerate elliptic problem from subsonic-sonic flows in general nozzles, J. Differ. Equ., 267 (2019), 3778-3796.  doi: 10.1016/j.jde.2019.04.026.

[20]

C. J. Xie and Z. P. Xin, Global subsonic and subsonic-sonic flows through infinitely long nozzles, Indiana U. Math. J., 56 (2007), 2991-3023.  doi: 10.1512/iumj.2007.56.3108.

[21]

C. J. Xie and Z. P. Xin, Existence of global steady subsonic Euler flows through infinitely long nozzles, SIAM J. Math. Anal., 42 (2010), 751-784.  doi: 10.1137/09076667X.

[22]

J. X. Yin and C. P. Wang, Evolutionary weighted $p$-Laplacian with boundary degeneracy, J. Differ. Equ., 237 (2007), 421-445.  doi: 10.1016/j.jde.2007.03.012.

[1]

Mingjun Zhou, Jingxue Yin. Continuous subsonic-sonic flows in a two-dimensional semi-infinitely long nozzle. Electronic Research Archive, 2021, 29 (3) : 2417-2444. doi: 10.3934/era.2020122
[2]

Volker Elling. Subsonic irrotational inviscid flow around certain bodies with two protruding corners. Communications on Pure and Applied Analysis, 2018, 17 (2) : 539-555. doi: 10.3934/cpaa.2018029
[3]

Eun Heui Kim. Boundary gradient estimates for subsonic solutions of compressible transonic potential flows. Conference Publications, 2007, 2007 (Special) : 573-579. doi: 10.3934/proc.2007.2007.573
[4]

Chunpeng Wang. Boundary behavior and asymptotic behavior of solutions to a class of parabolic equations with boundary degeneracy. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 1041-1060. doi: 10.3934/dcds.2016.36.1041
[5]

Genni Fragnelli, Gisèle Ruiz Goldstein, Jerome Goldstein, Rosa Maria Mininni, Silvia Romanelli. Generalized Wentzell boundary conditions for second order operators with interior degeneracy. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 697-715. doi: 10.3934/dcdss.2016023
[6]

Hayk Mikayelyan, Henrik Shahgholian. Convexity of the free boundary for an exterior free boundary problem involving the perimeter. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1431-1443. doi: 10.3934/cpaa.2013.12.1431
[7]

Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10
[8]

Avner Friedman. Free boundary problems arising in biology. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 193-202. doi: 10.3934/dcdsb.2018013
[9]

Yang Zhang. A free boundary problem of the cancer invasion. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1323-1343. doi: 10.3934/dcdsb.2021092
[10]

Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003
[11]

Xinfu Chen, Huibin Cheng. Regularity of the free boundary for the American put option. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1751-1759. doi: 10.3934/dcdsb.2012.17.1751
[12]

Jia-Feng Cao, Wan-Tong Li, Fei-Ying Yang. Dynamics of a nonlocal SIS epidemic model with free boundary. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 247-266. doi: 10.3934/dcdsb.2017013
[13]

Avner Friedman. Free boundary problems for systems of Stokes equations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1455-1468. doi: 10.3934/dcdsb.2016006
[14]

Tong Yang, Fahuai Yi. Global existence and uniqueness for a hyperbolic system with free boundary. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 763-780. doi: 10.3934/dcds.2001.7.763
[15]

Wenzhen Gan, Peng Zhou. A revisit to the diffusive logistic model with free boundary condition. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 837-847. doi: 10.3934/dcdsb.2016.21.837
[16]

Serena Dipierro, Enrico Valdinoci. (Non)local and (non)linear free boundary problems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (3) : 465-476. doi: 10.3934/dcdss.2018025
[17]

Naoki Sato, Toyohiko Aiki, Yusuke Murase, Ken Shirakawa. A one dimensional free boundary problem for adsorption phenomena. Networks and Heterogeneous Media, 2014, 9 (4) : 655-668. doi: 10.3934/nhm.2014.9.655
[18]

Yongzhi Xu. A free boundary problem model of ductal carcinoma in situ. Discrete and Continuous Dynamical Systems - B, 2004, 4 (1) : 337-348. doi: 10.3934/dcdsb.2004.4.337
[19]

Carlos E. Kenig, Tatiana Toro. On the free boundary regularity theorem of Alt and Caffarelli. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 397-422. doi: 10.3934/dcds.2004.10.397
[20]

Gary Bunting, Yihong Du, Krzysztof Krakowski. Spreading speed revisited: Analysis of a free boundary model. Networks and Heterogeneous Media, 2012, 7 (4) : 583-603. doi: 10.3934/nhm.2012.7.583

2021 Impact Factor: 1.273

Tools

Metrics

  • PDF downloads (193)
  • HTML views (199)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy