American Institute of Mathematical Sciences

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doi: 10.3934/era.2020122

Continuous subsonic-sonic flows in a two-dimensional semi-infinitely long nozzle

Mingjun Zhou 1,, and  Jingxue Yin 2,

1. 

School of Mathematics, Jilin University, Changchun 130012, China
2. 

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

* Corresponding author: Mingjun Zhou

Received  August 2020 Revised  October 2020 Published  November 2020

Fund Project: Mingjun Zhou is supported by the National Natural Science Foundation of China (Grant Nos. 11925105 and 12001227), Jingxue Yin is supported by the National Natural Science Foundation of China Grant No. 11771156, Guangdong Basic and Applied Basic Research Foundation Grant No. 2020B1515310013, Science and Technology Program of Guangzhou No. 2019050001, and the Natural Science Foundation of Guangzhou Grant No. 201804010391

This paper focuses on two-dimensional continuous subsonic-sonic potential flows in a semi-infinitely long nozzle with a straight lower wall and an upper wall which is convergent at the outlet while straight at the far fields. It is proved that if the variation rate of the cross section of the nozzle is suitably small, there exists a unique continuous subsonic-sonic flows in the nozzle such that the sonic curve intersects the upper wall at a fixed point and the velocity of the flow is along the normal direction at the sonic curve. Furthermore, the sonic curve is free, where the flow is singular in the sense that the flow speed is only Hölder continuous and the flow acceleration blows up. Additionally, the asymptotic behaviors of the flow speed at the far fields is shown.

Keywords: Degeneracy, free boundary, continuous subsonic-sonic flow, nonlocal boundary condition.
Mathematics Subject Classification: 35R35, 35Q35, 76N10, 35J70.
Citation: Mingjun Zhou, Jingxue Yin. Continuous subsonic-sonic flows in a two-dimensional semi-infinitely long nozzle. Electronic Research Archive, doi: 10.3934/era.2020122
References:
[1]

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

[2]

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

[3]

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

[4]

C. Chen, L. Du, C. Xie and Z. 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.  Google Scholar

[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.  Google Scholar

[6]

G.-Q. Chen, F.-M. Huang, T.-Y. Wang and W. Xiang, Steady Euler flows with large vorticity and characteristic discontinuities in arbitrary infinitely long nozzles, Adv. Math., 346 (2019), 946-1008.  doi: 10.1016/j.aim.2019.02.002.  Google Scholar

[7]

C. Chen and C. Xie, Existence of steady subsonic Euler flows through infinitely long periodic nozzles, J. Differential Equations, 252 (2012), 4315-4331.  doi: 10.1016/j.jde.2011.12.015.  Google Scholar

[8]

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, N. Y., 1948.  Google Scholar

[9]

G. Dong, Nonlinear Partial Differential Equations of Second Order, Translations of Mathematical Monographs. 95, American Mathematical Society, Providence, RI, 1991. doi: 10.1090/mmono/095.  Google Scholar

[10]

L. Du and B. Duan, Subsonic Euler flows with large vorticity through an infinitely long axisymmetric nozzle, J. Math. Fluid Mech., 18 (2016), 511-530.  doi: 10.1007/s00021-016-0255-8.  Google Scholar

[11]

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

[12]

L. Du, Z. 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.  Google Scholar

[13]

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.  Google Scholar

[14]

F. Huang, T. Wang and Y. Wang, On multi-dimensional sonic-subsonic flow, Acta Math. Sci. Ser. B (Engl. Ed.), 31 (2011), 2131-2140.  doi: 10.1016/S0252-9602(11)60389-5.  Google Scholar

[15]

A. Kuz'min, Boundary Value Problems for Transonic Flow, John Wiley and Sons, Ltd, 2002. Google Scholar

[16]

Y. Nie and C. 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.  Google Scholar

[17]

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

[18]

C. Wang, Continuous subsonic-sonic flows in a general nozzle, J. Differential Equations, 259 (2015), 2546-2575.  doi: 10.1016/j.jde.2015.03.036.  Google Scholar

[19]

C. Wang, A free boundary problem of a degenerate elliptic equation and subsonic-sonic flows with general sonic curves, SIAM J. Math. Anal., 51 (2019), 4977-5010.  doi: 10.1137/19M1255860.  Google Scholar

[20]

C. Wang and Z. 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.  Google Scholar

[21]

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

[22]

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

[23]

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

[24]

C. Xie and Z. Xin, Global subsonic and subsonic-sonic flows through infinitely long axially symmetric nozzles, J. Differential Equations, 248 (2010), 2657-2683.  doi: 10.1016/j.jde.2010.02.007.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

C. Chen, L. Du, C. Xie and Z. 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.  Google Scholar

[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.  Google Scholar

[6]

G.-Q. Chen, F.-M. Huang, T.-Y. Wang and W. Xiang, Steady Euler flows with large vorticity and characteristic discontinuities in arbitrary infinitely long nozzles, Adv. Math., 346 (2019), 946-1008.  doi: 10.1016/j.aim.2019.02.002.  Google Scholar

[7]

C. Chen and C. Xie, Existence of steady subsonic Euler flows through infinitely long periodic nozzles, J. Differential Equations, 252 (2012), 4315-4331.  doi: 10.1016/j.jde.2011.12.015.  Google Scholar

[8]

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, N. Y., 1948.  Google Scholar

[9]

G. Dong, Nonlinear Partial Differential Equations of Second Order, Translations of Mathematical Monographs. 95, American Mathematical Society, Providence, RI, 1991. doi: 10.1090/mmono/095.  Google Scholar

[10]

L. Du and B. Duan, Subsonic Euler flows with large vorticity through an infinitely long axisymmetric nozzle, J. Math. Fluid Mech., 18 (2016), 511-530.  doi: 10.1007/s00021-016-0255-8.  Google Scholar

[11]

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

[12]

L. Du, Z. 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.  Google Scholar

[13]

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.  Google Scholar

[14]

F. Huang, T. Wang and Y. Wang, On multi-dimensional sonic-subsonic flow, Acta Math. Sci. Ser. B (Engl. Ed.), 31 (2011), 2131-2140.  doi: 10.1016/S0252-9602(11)60389-5.  Google Scholar

[15]

A. Kuz'min, Boundary Value Problems for Transonic Flow, John Wiley and Sons, Ltd, 2002. Google Scholar

[16]

Y. Nie and C. 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.  Google Scholar

[17]

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

[18]

C. Wang, Continuous subsonic-sonic flows in a general nozzle, J. Differential Equations, 259 (2015), 2546-2575.  doi: 10.1016/j.jde.2015.03.036.  Google Scholar

[19]

C. Wang, A free boundary problem of a degenerate elliptic equation and subsonic-sonic flows with general sonic curves, SIAM J. Math. Anal., 51 (2019), 4977-5010.  doi: 10.1137/19M1255860.  Google Scholar

[20]

C. Wang and Z. 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.  Google Scholar

[21]

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

[22]

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

[23]

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

[24]

C. Xie and Z. Xin, Global subsonic and subsonic-sonic flows through infinitely long axially symmetric nozzles, J. Differential Equations, 248 (2010), 2657-2683.  doi: 10.1016/j.jde.2010.02.007.  Google Scholar

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