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Uniqueness of steady 1-D shock solutions in a finite nozzle via vanishing viscosity aguments
A degenerate elliptic problem from subsonic-sonic flows in convergent nozzles
School of Mathematics, Jilin University, Changchun 130012, Jilin, China |
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.
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. |
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