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July & August  2021, 20(7&8): 2555-2577. doi: 10.3934/cpaa.2021070

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

 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.

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.

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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.
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