doi: 10.3934/cpaa.2021066

Uniqueness of steady 1-D shock solutions in a finite nozzle via vanishing viscosity aguments

1. 

School of Mathematical Sciences, MOE-LSC, and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240, China

2. 

Department of Mathematics, School of Science, Wuhan University of Technology, Wuhan 430070, China

* Corresponding author

Dedicated to Professor Shuxing Chen on the occasion of his 80th birthday

Received  February 2021 Revised  March 2021 Published  April 2021

Fund Project: The research was supported by Natural Science Foundation of China under Grant Nos. 11631008 and 11971308

This paper studies the uniqueness of steady 1-D shock solutions in a finite flat nozzle via vanishing viscosity arguments. It is proved that, for both barotropic gases and non-isentropic gases, the steady viscous shock solutions converge under the $ \mathcal{L}^{1} $ norm. Hence only one shock solution of the inviscid Euler system could be the limit as the viscosity coefficient goes to $ 0 $, which shows the uniqueness of the steady 1-D shock solution in a finite flat nozzle. Moreover, the position of the shock front for the limit shock solution is also obtained.

Citation: Beixiang Fang, Qin Zhao. Uniqueness of steady 1-D shock solutions in a finite nozzle via vanishing viscosity aguments. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021066
References:
[1]

B. Barker, B. Melinand and K. Zumbrun, Existence and stability of steady noncharacteristic solutions on a finite interval of full compressible Navier-Stokes equations, preprint, arXiv: 1911.06691. Google Scholar

[2]

S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. Math., 161 (2005), 223-342.  doi: 10.4007/annals.2005.161.223.  Google Scholar

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G. Q. ChenJ. Chen and K. Song, Transonic nozzle flows and free boundary problems for the full Euler equations, J. Differ. Equ., 229 (2006), 92-120.  doi: 10.1016/j.jde.2006.04.015.  Google Scholar

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G. Q. Chen and M. Feldman, Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type, J. Amer. Math. Soc., 16 (2003), 461-494.  doi: 10.1090/S0894-0347-03-00422-3.  Google Scholar

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G. Q. Chen and H. Yuan, Local uniqueness of steady spherical transonic shock-fronts for the three-dimensional full Euler equations, Commun. Pure Appl. Anal., 12 (2013), 2515-2542.  doi: 10.3934/cpaa.2013.12.2515.  Google Scholar

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S. Chen, Compressible flow and transonic shock in a diverging nozzle, Commun. Math. Phys., 289 (2009), 75-106.  doi: 10.1007/s00220-009-0811-7.  Google Scholar

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R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Springer-Verlag, New York, 1948.  Google Scholar

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D. Cui and H. Yin, The uniqueness of a transonic shock in a nozzle for the 2-D complete Euler system with the variable end pressure, J. Partial Differ. Equ., 21 (2008), 263-288.   Google Scholar

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P. EmbidJ. Goodman and A. Majda, Multiple steady states for 1-D transonic flow, SIAM J. Sci. Statist. Comput., 5 (1984), 21-41.  doi: 10.1137/0905002.  Google Scholar

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B. Fang and Z. Xin, On admissible locations of transonic shock fronts for steady Euler flows in an almost flat finite nozzle with prescribed receiver pressure, Commun. Pure Appl. Math., online (2020). doi: 10.1002/cpa.21966.  Google Scholar

[11]

D. Gilbarg, The existence and limit behavior of the one-dimensional shock layer, Amer. J. Math., 73 (1951), 256-274.  doi: 10.2307/2372177.  Google Scholar

[12]

J. Goodman and Z. Xin, Viscous limits for piecewise smooth solutions to systems of conservation laws, Arch. Ration. Mech. Anal., 121 (1992), 235-265.  doi: 10.1007/BF00410614.  Google Scholar

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O. Guès and M. Williams, Curved shocks as viscous limits: a boundary problem approach, Indiana Univ. Math. J., 51 (2002), 421-450.  doi: 10.1512/iumj.2002.51.2197.  Google Scholar

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O. GuèsG. MétivierM. Williams and K. Zumbrun, Multidimensional viscous shocks Ⅱ: The small viscosity limit, Commun. Pure Appl. Math., 57 (2004), 141-218.  doi: 10.1002/cpa.10115.  Google Scholar

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J. LiZ. Xin and H. Yin, On transonic shocks in a nozzle with variable end pressures, Commun. Math. Phys., 291 (2009), 111-150.  doi: 10.1007/s00220-009-0870-9.  Google Scholar

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J. LiZ. Xin and H. Yin, Monotonicity and uniqueness of a 3D transonic shock solution in a conic nozzle with variable end pressure, Pacific J. Math., 254 (2011), 129-171.  doi: 10.2140/pjm.2011.254.129.  Google Scholar

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J. LiZ. Xin and H. Yin, Transonic shocks for the full compressible Euler system in a general two-dimensional de Laval nozzle, Arch. Ration. Mech. Anal., 207 (2013), 533-581.  doi: 10.1007/s00205-012-0580-x.  Google Scholar

[19]

T. P. Liu, Transonic gas flow in a duct of varying area, Arch. Rational Mech. Anal., 80 (1982), 1-18.  doi: 10.1007/BF00251521.  Google Scholar

[20]

T. P. Liu, Nonlinear stability and instability of transonic flows through a nozzle, Commun. Math. Phys., 83 (1982), 243-260.  doi: 10.1007/BF01976043.  Google Scholar

[21]

B. Melinand and K. Zumbrun, Existence and stability of steady compressible Navier-Stokes solutions on a finite interval with noncharacteristic boundary conditions, Phys. D, 394 (2019), 16-25.  doi: 10.1016/j.physd.2019.01.006.  Google Scholar

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M. Strani, Existence and uniqueness of a positive connection for the scalar viscous shallow water system in a bounded interval, Commun. Pure Appl. Anal., 13 (2014), 1653-1667.  doi: 10.3934/cpaa.2014.13.1653.  Google Scholar

[23]

Y. Wang, Zero dissipation limit of the compressible heat-conducting Navier-Stokes equations in the presence of the shock, Acta Math. Sci. Ser. B., 28 (2008), 727-748.  doi: 10.1016/S0252-9602(08)60074-0.  Google Scholar

[24]

Z. Xin and H. Yin, Transonic shock in a nozzle Ⅰ: Two-dimensional case, Commun. Pure Appl. Math., 58 (2005), 999-1050.  doi: 10.1002/cpa.20025.  Google Scholar

show all references

References:
[1]

B. Barker, B. Melinand and K. Zumbrun, Existence and stability of steady noncharacteristic solutions on a finite interval of full compressible Navier-Stokes equations, preprint, arXiv: 1911.06691. Google Scholar

[2]

S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. Math., 161 (2005), 223-342.  doi: 10.4007/annals.2005.161.223.  Google Scholar

[3]

G. Q. ChenJ. Chen and K. Song, Transonic nozzle flows and free boundary problems for the full Euler equations, J. Differ. Equ., 229 (2006), 92-120.  doi: 10.1016/j.jde.2006.04.015.  Google Scholar

[4]

G. Q. Chen and M. Feldman, Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type, J. Amer. Math. Soc., 16 (2003), 461-494.  doi: 10.1090/S0894-0347-03-00422-3.  Google Scholar

[5]

G. Q. Chen and H. Yuan, Local uniqueness of steady spherical transonic shock-fronts for the three-dimensional full Euler equations, Commun. Pure Appl. Anal., 12 (2013), 2515-2542.  doi: 10.3934/cpaa.2013.12.2515.  Google Scholar

[6]

S. Chen, Compressible flow and transonic shock in a diverging nozzle, Commun. Math. Phys., 289 (2009), 75-106.  doi: 10.1007/s00220-009-0811-7.  Google Scholar

[7]

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Springer-Verlag, New York, 1948.  Google Scholar

[8]

D. Cui and H. Yin, The uniqueness of a transonic shock in a nozzle for the 2-D complete Euler system with the variable end pressure, J. Partial Differ. Equ., 21 (2008), 263-288.   Google Scholar

[9]

P. EmbidJ. Goodman and A. Majda, Multiple steady states for 1-D transonic flow, SIAM J. Sci. Statist. Comput., 5 (1984), 21-41.  doi: 10.1137/0905002.  Google Scholar

[10]

B. Fang and Z. Xin, On admissible locations of transonic shock fronts for steady Euler flows in an almost flat finite nozzle with prescribed receiver pressure, Commun. Pure Appl. Math., online (2020). doi: 10.1002/cpa.21966.  Google Scholar

[11]

D. Gilbarg, The existence and limit behavior of the one-dimensional shock layer, Amer. J. Math., 73 (1951), 256-274.  doi: 10.2307/2372177.  Google Scholar

[12]

J. Goodman and Z. Xin, Viscous limits for piecewise smooth solutions to systems of conservation laws, Arch. Ration. Mech. Anal., 121 (1992), 235-265.  doi: 10.1007/BF00410614.  Google Scholar

[13]

O. Guès and M. Williams, Curved shocks as viscous limits: a boundary problem approach, Indiana Univ. Math. J., 51 (2002), 421-450.  doi: 10.1512/iumj.2002.51.2197.  Google Scholar

[14]

O. GuèsG. MétivierM. Williams and K. Zumbrun, Multidimensional viscous shocks Ⅱ: The small viscosity limit, Commun. Pure Appl. Math., 57 (2004), 141-218.  doi: 10.1002/cpa.10115.  Google Scholar

[15]

D. Hoff and T. P. Liu, The inviscid limit for the Navier-Stokes equations of compressible, isentropic flow with shock data, Indiana Univ. Math. J., 38 (1989), 861-915.  doi: 10.1512/iumj.1989.38.38041.  Google Scholar

[16]

J. LiZ. Xin and H. Yin, On transonic shocks in a nozzle with variable end pressures, Commun. Math. Phys., 291 (2009), 111-150.  doi: 10.1007/s00220-009-0870-9.  Google Scholar

[17]

J. LiZ. Xin and H. Yin, Monotonicity and uniqueness of a 3D transonic shock solution in a conic nozzle with variable end pressure, Pacific J. Math., 254 (2011), 129-171.  doi: 10.2140/pjm.2011.254.129.  Google Scholar

[18]

J. LiZ. Xin and H. Yin, Transonic shocks for the full compressible Euler system in a general two-dimensional de Laval nozzle, Arch. Ration. Mech. Anal., 207 (2013), 533-581.  doi: 10.1007/s00205-012-0580-x.  Google Scholar

[19]

T. P. Liu, Transonic gas flow in a duct of varying area, Arch. Rational Mech. Anal., 80 (1982), 1-18.  doi: 10.1007/BF00251521.  Google Scholar

[20]

T. P. Liu, Nonlinear stability and instability of transonic flows through a nozzle, Commun. Math. Phys., 83 (1982), 243-260.  doi: 10.1007/BF01976043.  Google Scholar

[21]

B. Melinand and K. Zumbrun, Existence and stability of steady compressible Navier-Stokes solutions on a finite interval with noncharacteristic boundary conditions, Phys. D, 394 (2019), 16-25.  doi: 10.1016/j.physd.2019.01.006.  Google Scholar

[22]

M. Strani, Existence and uniqueness of a positive connection for the scalar viscous shallow water system in a bounded interval, Commun. Pure Appl. Anal., 13 (2014), 1653-1667.  doi: 10.3934/cpaa.2014.13.1653.  Google Scholar

[23]

Y. Wang, Zero dissipation limit of the compressible heat-conducting Navier-Stokes equations in the presence of the shock, Acta Math. Sci. Ser. B., 28 (2008), 727-748.  doi: 10.1016/S0252-9602(08)60074-0.  Google Scholar

[24]

Z. Xin and H. Yin, Transonic shock in a nozzle Ⅰ: Two-dimensional case, Commun. Pure Appl. Math., 58 (2005), 999-1050.  doi: 10.1002/cpa.20025.  Google Scholar

Figure 1.  Steady normal shocks in a flat nozzle
Figure 2.  The velocity functions $ u^{\varepsilon}(x) $ for different viscosity $ \varepsilon>0 $ and their limit as $ \varepsilon \to 0+ $
Figure 3.  The graph of the function $ f(u) $
Figure 4.  The auxilliary functions for $ f(u) $
Figure 5.  Auxilliary points for $ f(u) $
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