American Institute of Mathematical Sciences

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doi: 10.3934/cpaa.2021015

On a supersonic-sonic patch arising from the frankl problem in transonic flows

Yanbo Hu 1, and  Jiequan Li 2,,

1. 

Department of Mathematics, Hangzhou Normal University, Hangzhou, 311121, China
2. 

Laboratory of Computational Physics, Institute of Applied Physics, and Computational Mathematics, Beijing, 100088, China
3. 

Center for Applied Physics and Technology, Peking University, 100871, China

* Corresponding author

Dedicated to the celebration of the 80th birthday of Professor Shuxing Chen

Received  September 2020 Revised  December 2020 Early access  February 2021

Fund Project: The first author is supported by the Zhejiang Provincial Natural Science Foundation (No. LY21A010017). The second author is supported by the Natural Science Foundation of China (Nos: 11771054, 91852207, 12072042), National Key Project(GJXM92579) and Foundation of LCP

We construct a supersonic-sonic smooth patch solution for the two dimensional steady Euler equations in gas dynamics. This patch is extracted from the Frankl problem in the study of transonic flow with local supersonic bubble over an airfoil. Based on the methodology of characteristic decompositions, we establish the global existence and regularity of solutions in a partial hodograph coordinate system in terms of angle variables. The original problem is solved by transforming the solution in the partial hodograph plane back to that in the physical plane. Moreover, the uniform regularity of the solution and the regularity of an associated sonic curve are also verified.

Keywords: Steady Euler equations, Frankl problem, supersonic-sonic solution, existence, characteristic decomposition.
Mathematics Subject Classification: Primary: 35Q31, 35Q35, 35L80, 76H05.
Citation: Yanbo Hu, Jiequan Li. On a supersonic-sonic patch arising from the frankl problem in transonic flows. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021015
References:
[1]

J. Barros-Neto and I. M. Gelfand, Fundamental solutions for the Tricomi operator, Ⅰ, Ⅱ, Ⅲ, Duke Math. J., 98 (1999), 465–483; 111 (2002), 561–584; 128 (2005), 119–140. doi: 10.1215/S0012-7094-02-11137-5.  Google Scholar

[2]

L. Bers, On the continuation of a potential gas flow across the sonic line, Tech. Notes Nat. Adv. Comm. Aeronaut., No. 2058, 1950.  Google Scholar

[3]

G. Q. Chen, C. Dafermos, M. Slemrod and D. H. Wang, On two-dimensional sonic-subsonic flow, Commun. Math. Phys., 271 (2007), 635-637.  doi: 10.1007/s00220-007-0211-9.  Google Scholar

[4]

G. Q. Chen, F. M. Huang and T. 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

[5]

S. X. Chen, Generalized Tricomi problem for a quasilinear mixed type equation, Chin. Ann. Math. Ser. B, 30 (2009), 527-538.  doi: 10.1007/s11401-009-0215-1.  Google Scholar

[6]

S. X. Chen, The fundamental solution of the Keldysh type operator, Sci. China Ser. A, 52 (2009), 1829-1843.  doi: 10.1007/s11425-009-0069-8.  Google Scholar

[7]

S. X. Chen, Tricomi problem for a mixed equation of second order with discontinuous coefficients, Acta Math. Sci. Ser. B, 29 (2009), 569-582.  doi: 10.1016/S0252-9602(09)60054-0.  Google Scholar

[8]

S. X. Chen, Mixed type equations in gas dynamics, Quart. Appl. Math., 68 (2010), 487-511.  doi: 10.1090/S0033-569X-2010-01164-9.  Google Scholar

[9]

S. X. Chen, A mixed equation of Tricomi-Keldysh type, J. Hyperbolic Differ. Equ., 9 (2012), 545-553.  doi: 10.1142/S0219891612500178.  Google Scholar

[10]

J. Cole and L. Cook, Transonic Aerodynamics, North-Holland Series in Applied Mathematics and Mechanics, Elsevier, Amsterdam, 1986. Google Scholar

[11]

L. Cook, A uniqueness proof for a transonic flow problem, Indiana Univ. Math. J., 27 (1978), 51-71.  doi: 10.1512/iumj.1978.27.27005.  Google Scholar

[12]

R. Courant and K. Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York, 1948.  Google Scholar

[13]

F. Frankl, On the formation of shock waves in subsonic flows with local supersonic velocities, Prikl. Mat. Mech., 11 (1947), 199-202.   Google Scholar

[14]

Y. B. Hu and J. Q. Li, Sonic-supersonic solutions for the two-dimensional steady full Euler equations, Arch. Ration. Mech. Anal., 235 (2020), 1819-1871.  doi: 10.1007/s00205-019-01454-w.  Google Scholar

[15]

Y. B. Hu and J. Q. Li, On a global supersonic-sonic patch characterized by 2-D steady full Euler equations, Adv. Differ. Equ., 25 (2020), 213-254.   Google Scholar

[16]

A. Kuz'min, Solvability of a problem for transonic flow with a local supersonic region, Nonlinear Differ. Equ. Appl., 8 (2001), 299-321.  doi: 10.1007/PL00001450.  Google Scholar

[17]

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

[18]

A. Kuz'min, A modified Frankl-Morawetz problem on a transonic flow past an airfoil, Differ. Equ., 40 (2004), 1455-1460.  doi: 10.1007/s10625-005-0077-6.  Google Scholar

[19]

J. Q. Li, Z. C. Yang and Y. X. Zheng, Characteristic decompositions and interactions of rarefaction waves of 2-D Euler equations, J. Differ. Equ., 250 (2011), 782-798.  doi: 10.1016/j.jde.2010.07.009.  Google Scholar

[20]

J. Q. Li and Y. X. Zheng, Interaction of rarefaction waves of the two-dimensional self-similar Euler equations, Arch. Ration. Mech. Anal., 193 (2009), 623-657.  doi: 10.1007/s00205-008-0140-6.  Google Scholar

[21]

M. J. Li and Y. X. Zheng, Semi-hyperbolic patches of solutions of the two-dimensional Euler equations, Arch. Ration. Mech. Anal., 201 (2011), 1069-1096.  doi: 10.1007/s00205-011-0410-6.  Google Scholar

[22]

C. Morawetz, A uniqueness theorem for Frankl's problem, Commun. Pure Appl. Math., 7 (1954), 697-703.  doi: 10.1002/cpa.3160070406.  Google Scholar

[23]

C. Morawetz, On the non-existence of continuous transonic flow past profiles Ⅰ, Ⅱ, Commun. Pure Appl. Math., 9 (1956), 45–68; 10 (1957), 107–131. doi: 10.1002/cpa.3160100105.  Google Scholar

[24]

C. Morawetz, On a weak solution for a transonic flow problem, Commun. Pure Appl. Math., 38 (1985), 797-817.  doi: 10.1002/cpa.3160380610.  Google Scholar

[25]

K. Song, Q. Wang and Y. X. Zheng, The regularity of semihyperbolic patches near sonic lines for the 2-D Euler system in gas dynamics, SIAM J. Math. Anal., 47 (2015), 2200-2219.  doi: 10.1137/140964382.  Google Scholar

[26]

R. Vaglio-Laurin, Transonic rotational flow over a convex corner, J. Fluid Mech., 9 (1960), 81-103.  doi: 10.1017/S0022112060000931.  Google Scholar

[27]

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.  Google Scholar

[28]

C. J. Xie and Z. P. 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

[29]

T. Y. Zhang and Y. X. Zheng, Sonic-supersonic solutions for the steady Euler equations, Indiana Univ. Math. J., 63 (2014), 1785-1817.  doi: 10.1512/iumj.2014.63.5434.  Google Scholar

show all references

References:
[1]

J. Barros-Neto and I. M. Gelfand, Fundamental solutions for the Tricomi operator, Ⅰ, Ⅱ, Ⅲ, Duke Math. J., 98 (1999), 465–483; 111 (2002), 561–584; 128 (2005), 119–140. doi: 10.1215/S0012-7094-02-11137-5.  Google Scholar

[2]

L. Bers, On the continuation of a potential gas flow across the sonic line, Tech. Notes Nat. Adv. Comm. Aeronaut., No. 2058, 1950.  Google Scholar

[3]

G. Q. Chen, C. Dafermos, M. Slemrod and D. H. Wang, On two-dimensional sonic-subsonic flow, Commun. Math. Phys., 271 (2007), 635-637.  doi: 10.1007/s00220-007-0211-9.  Google Scholar

[4]

G. Q. Chen, F. M. Huang and T. 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

[5]

S. X. Chen, Generalized Tricomi problem for a quasilinear mixed type equation, Chin. Ann. Math. Ser. B, 30 (2009), 527-538.  doi: 10.1007/s11401-009-0215-1.  Google Scholar

[6]

S. X. Chen, The fundamental solution of the Keldysh type operator, Sci. China Ser. A, 52 (2009), 1829-1843.  doi: 10.1007/s11425-009-0069-8.  Google Scholar

[7]

S. X. Chen, Tricomi problem for a mixed equation of second order with discontinuous coefficients, Acta Math. Sci. Ser. B, 29 (2009), 569-582.  doi: 10.1016/S0252-9602(09)60054-0.  Google Scholar

[8]

S. X. Chen, Mixed type equations in gas dynamics, Quart. Appl. Math., 68 (2010), 487-511.  doi: 10.1090/S0033-569X-2010-01164-9.  Google Scholar

[9]

S. X. Chen, A mixed equation of Tricomi-Keldysh type, J. Hyperbolic Differ. Equ., 9 (2012), 545-553.  doi: 10.1142/S0219891612500178.  Google Scholar

[10]

J. Cole and L. Cook, Transonic Aerodynamics, North-Holland Series in Applied Mathematics and Mechanics, Elsevier, Amsterdam, 1986. Google Scholar

[11]

L. Cook, A uniqueness proof for a transonic flow problem, Indiana Univ. Math. J., 27 (1978), 51-71.  doi: 10.1512/iumj.1978.27.27005.  Google Scholar

[12]

R. Courant and K. Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York, 1948.  Google Scholar

[13]

F. Frankl, On the formation of shock waves in subsonic flows with local supersonic velocities, Prikl. Mat. Mech., 11 (1947), 199-202.   Google Scholar

[14]

Y. B. Hu and J. Q. Li, Sonic-supersonic solutions for the two-dimensional steady full Euler equations, Arch. Ration. Mech. Anal., 235 (2020), 1819-1871.  doi: 10.1007/s00205-019-01454-w.  Google Scholar

[15]

Y. B. Hu and J. Q. Li, On a global supersonic-sonic patch characterized by 2-D steady full Euler equations, Adv. Differ. Equ., 25 (2020), 213-254.   Google Scholar

[16]

A. Kuz'min, Solvability of a problem for transonic flow with a local supersonic region, Nonlinear Differ. Equ. Appl., 8 (2001), 299-321.  doi: 10.1007/PL00001450.  Google Scholar

[17]

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

[18]

A. Kuz'min, A modified Frankl-Morawetz problem on a transonic flow past an airfoil, Differ. Equ., 40 (2004), 1455-1460.  doi: 10.1007/s10625-005-0077-6.  Google Scholar

[19]

J. Q. Li, Z. C. Yang and Y. X. Zheng, Characteristic decompositions and interactions of rarefaction waves of 2-D Euler equations, J. Differ. Equ., 250 (2011), 782-798.  doi: 10.1016/j.jde.2010.07.009.  Google Scholar

[20]

J. Q. Li and Y. X. Zheng, Interaction of rarefaction waves of the two-dimensional self-similar Euler equations, Arch. Ration. Mech. Anal., 193 (2009), 623-657.  doi: 10.1007/s00205-008-0140-6.  Google Scholar

[21]

M. J. Li and Y. X. Zheng, Semi-hyperbolic patches of solutions of the two-dimensional Euler equations, Arch. Ration. Mech. Anal., 201 (2011), 1069-1096.  doi: 10.1007/s00205-011-0410-6.  Google Scholar

[22]

C. Morawetz, A uniqueness theorem for Frankl's problem, Commun. Pure Appl. Math., 7 (1954), 697-703.  doi: 10.1002/cpa.3160070406.  Google Scholar

[23]

C. Morawetz, On the non-existence of continuous transonic flow past profiles Ⅰ, Ⅱ, Commun. Pure Appl. Math., 9 (1956), 45–68; 10 (1957), 107–131. doi: 10.1002/cpa.3160100105.  Google Scholar

[24]

C. Morawetz, On a weak solution for a transonic flow problem, Commun. Pure Appl. Math., 38 (1985), 797-817.  doi: 10.1002/cpa.3160380610.  Google Scholar

[25]

K. Song, Q. Wang and Y. X. Zheng, The regularity of semihyperbolic patches near sonic lines for the 2-D Euler system in gas dynamics, SIAM J. Math. Anal., 47 (2015), 2200-2219.  doi: 10.1137/140964382.  Google Scholar

[26]

R. Vaglio-Laurin, Transonic rotational flow over a convex corner, J. Fluid Mech., 9 (1960), 81-103.  doi: 10.1017/S0022112060000931.  Google Scholar

[27]

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.  Google Scholar

[28]

C. J. Xie and Z. P. 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

[29]

T. Y. Zhang and Y. X. Zheng, Sonic-supersonic solutions for the steady Euler equations, Indiana Univ. Math. J., 63 (2014), 1785-1817.  doi: 10.1512/iumj.2014.63.5434.  Google Scholar

Figure 1.  Transonic phenomena in a duct
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Figure 2.  The modified Frankl problem. With a velocity distribution on the arcs $ \widehat{PE} $ and $ \widehat{FQ} $, find an airfoil's arc $ \widehat{EF} $, free of boundary conditions, for the correctness of the problem in the class of smooth solutions
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Figure 3.  The region $ \Omega_ \varepsilon $
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