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June  2020, 19(6): 3387-3428. doi: 10.3934/cpaa.2020150

Global solutions of shock reflection problem for the pressure gradient system

Hanchun Yang , Meimei Zhang and  Qin Wang ,

Department of Mathematics and Statistics, Yunnan University, Kunming, 650091, China

* Corresponding author

Received  December 2019 Revised  December 2019 Published  March 2020

Fund Project: The first author is supported by NSF of Yunnan University (No. 2019FY003007). The third author is supported by National Natural Science Foundation of China (No. 11761077) and Key Project of Yunnan Provincial Science and Technology Department and Yunnan University (No. 2018FY001(-014))

We are concerned with the shock reflection in gas dynamics for the pressure gradient system. Experimental and computational analysis has shown that two patterns of regular reflection may occur: supersonic and subsonic reflection. In this paper we establish the global existence of solutions for both configurations. The ideas and techniques developed here will be useful for the two-dimensional Riemann problems for hyperbolic conservation laws.

Keywords: Pressure gradient system, von Neumann conjecture, shock reflection, free boundary, sonic boundary, hyperbolic-elliptic mixed type, Leray-Schauder degree theory, existence, regularity.
Mathematics Subject Classification: Primary: 35L50, 35L67, 76H05; Secondary: 35J67.
Citation: Hanchun Yang, Meimei Zhang, Qin Wang. Global solutions of shock reflection problem for the pressure gradient system. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3387-3428. doi: 10.3934/cpaa.2020150
References:
[1]

M. Bae, G. Q. Chen and M. Feldman, Regularity of solutions to regular shock reflection for potential flow, Invent. math., 175 (2009), 505-543.  doi: 10.1007/s00222-008-0156-4.  Google Scholar

[2]

G. Ben-Dor, Shock Wave Reflection Phenomena, Springer-Verlag, NewYork, 2007.  Google Scholar

[3]

M. Brio and J. K. Hunter, Mach reflection for the two-dimensional burgers equation, Physica D, 60 (1992), 194-207.  doi: 10.1016/0167-2789(92)90236-G.  Google Scholar

[4]

S. Čanić, B. L. Keyfitz and E. H. Kim, Free boundary problems for the unsteady transonic small disturbance equation: Transonic regular reflection, Meth. Appl. Anal., 7 (2000), 313-336.  doi: 10.4310/MAA.2000.v7.n2.a4.  Google Scholar

[5]

S. Čanić, B. L. Keyfitz and E. H. Kim, A free boundary problem for a quasi-linear degenerate elliptic equation: Regular reflection of weak shocks, Commun. Pure Appl. Math., 55 (2002), 71-92.  doi: 10.1002/cpa.10013.  Google Scholar

[6]

G. Q. Chen, X. M. Deng and W. Xiang, Shock diffraction by convex cornered wedges for the nonlinear wave system, Arch. Rational Mech. Anal., 211 (2014), 61-112.  doi: 10.1007/s00205-013-0681-1.  Google Scholar

[7]

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

[8]

G. Q. Chen and M. Feldman, Global solutions of shock reflection by large-angle wedges for potential flow, Ann. Math., 171 (2010), 1067-1182.  doi: 10.4007/annals.2010.171.1067.  Google Scholar

[9] G. Q. Chen and M. Feldman, The Mathematics of Shock Reflection-diffraction and Von Neumann's Conjectures, Research Monograpy, Annals of Mathematics Studies, Vol. 197, Princeton University Press, 2018.   Google Scholar
[10]

S. X. Chen, Linear approximation of shock reflection at a wedge with large angle, Commun. Partial Differ. Equ., 21 (1996), 1103-1118.  doi: 10.1080/03605309608821219.  Google Scholar

[11]

S. X. Chen, Study on mach reflection and mach configuration, Proc. Sympos. Appl. Math., 67 (2009), 53-71.  doi: 10.1090/psapm/067.1/2605212.  Google Scholar

[12]

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

[13]

V. Elling and T. P. Liu, Supersonic flow onto a solid wedge, Commun. Pure Appl. Math., 61 (2008), 1347-1448.  doi: 10.1002/cpa.20231.  Google Scholar

[14]

J. Glimm and A. J. Majda, Multidimensional Hyperbolic Problems and Computations, Springer-Verlag, NewYork, 1991. doi: 10.1007/978-1-4613-9121-0_8.  Google Scholar

[15]

Q. Han, Nonlinear Elliptic Equations of the Second Order, American Mathematical Society, Providence, RI, 2016.  Google Scholar

[16]

E. Harabetian, Diffraction of a weak shock by a wedge, Commun. Pure. Appl. Math., 40 (1987), 849-863.  doi: 10.1002/cpa.3160400608.  Google Scholar

[17]

H. Hornung, Regular and mach reflection of shock waves, Annu. Rev. Fluid Mech., 18 (1986), 33-58.   Google Scholar

[18]

J. K. Hunter, Transverse diffraction of nonlinear waves and singular rays, SIAM J. Appl. Math., 48 (1988), 1-37.  doi: 10.1137/0148001.  Google Scholar

[19]

J. K. Hunter and A. M. Tesdall, Weak shock reflection, in A celebration of Mathematical Modeling, Springer, (2004), 93–112.  Google Scholar

[20]

J. B. Keller and A. Blank, Diffraction and reflection of pulses by wedges and corners, Commun. Pure Appl. Math., 4 (1951), 75-94.  doi: 10.1002/cpa.3160040109.  Google Scholar

[21]

E. H. Kim, A global subsonic solution to an interacting transonic shock for the self-similar nonlinear wave equation, J. Differ. Equ., 248 (2010), 2906-2930.  doi: 10.1016/j.jde.2010.02.021.  Google Scholar

[22]

J. Q. Li, T. Zhang and S. L. Yang, The Two-dimensional Riemann Problem in Gas Dynamics, Pitman Monographs, Vol. 98, 1998.  Google Scholar

[23]

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: 10.2307/2000717.  Google Scholar

[24]

G. M. Lieberman, Oblique Derivative Problems for Elliptic Equations, World Scientific, 2013. doi: 10.1142/8679.  Google Scholar

[25]

E. Mach, Uber den verlauf von funkenwellen in der ebene und im raume, Sitzungsbr. Akad. Wiss. Wien, 78 (1878), 819-838.   Google Scholar

[26]

C. S. Morawetz, Potential theory for regular and mach reflection of a shock at a wedge, Commun. Pure Appl. Math., 47 (1994), 593-624.  doi: 10.1002/cpa.3160470502.  Google Scholar

[27]

M. Rigby, Transonic Shock Waves and Free Boundary Problems for the Nonlinear Wave System, PhD thesis, University of Oxford, 2018. Google Scholar

[28]

D. Serre, Shock Reflection in Gas Dynamics, Handbook of mathematical fluid dynamics, Elsevier, 2007.  Google Scholar

[29]

N. S. Trudinger, On an interpolation inequality and its applications to nonlinear elliptic equations, Proc. Amer. Math. Soc., 95 (1985), 73-78.  doi: 10.2307/2045576.  Google Scholar

[30]

J. von Neumann, Oblique Reflection of Shocks, Bureau of Ordinance, Explosives Research Report, 1943. Google Scholar

[31]

E. Zeidler, Nonlinear Functional Analysis and its Applications I, Springer-Verlag, NewYork, 1986. doi: 10.1007/978-1-4612-4838-5.  Google Scholar

[32]

Y. X. Zheng, Two-dimensional regular shock reflection for the pressure gradient system of conservation laws, Acta Math. Appl. Sin. Engl. Ser., 22 (2006), 177-210.  doi: 10.1007/s10255-006-0296-5.  Google Scholar

[33]

Y. X. Zheng, Systems of Conservation Laws: Two-dimensional Riemann Problems, Birkhäuser: Boston, 2001. doi: 10.1007/978-1-4612-0141-0.  Google Scholar

show all references

References:
[1]

M. Bae, G. Q. Chen and M. Feldman, Regularity of solutions to regular shock reflection for potential flow, Invent. math., 175 (2009), 505-543.  doi: 10.1007/s00222-008-0156-4.  Google Scholar

[2]

G. Ben-Dor, Shock Wave Reflection Phenomena, Springer-Verlag, NewYork, 2007.  Google Scholar

[3]

M. Brio and J. K. Hunter, Mach reflection for the two-dimensional burgers equation, Physica D, 60 (1992), 194-207.  doi: 10.1016/0167-2789(92)90236-G.  Google Scholar

[4]

S. Čanić, B. L. Keyfitz and E. H. Kim, Free boundary problems for the unsteady transonic small disturbance equation: Transonic regular reflection, Meth. Appl. Anal., 7 (2000), 313-336.  doi: 10.4310/MAA.2000.v7.n2.a4.  Google Scholar

[5]

S. Čanić, B. L. Keyfitz and E. H. Kim, A free boundary problem for a quasi-linear degenerate elliptic equation: Regular reflection of weak shocks, Commun. Pure Appl. Math., 55 (2002), 71-92.  doi: 10.1002/cpa.10013.  Google Scholar

[6]

G. Q. Chen, X. M. Deng and W. Xiang, Shock diffraction by convex cornered wedges for the nonlinear wave system, Arch. Rational Mech. Anal., 211 (2014), 61-112.  doi: 10.1007/s00205-013-0681-1.  Google Scholar

[7]

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

[8]

G. Q. Chen and M. Feldman, Global solutions of shock reflection by large-angle wedges for potential flow, Ann. Math., 171 (2010), 1067-1182.  doi: 10.4007/annals.2010.171.1067.  Google Scholar

[9] G. Q. Chen and M. Feldman, The Mathematics of Shock Reflection-diffraction and Von Neumann's Conjectures, Research Monograpy, Annals of Mathematics Studies, Vol. 197, Princeton University Press, 2018.   Google Scholar
[10]

S. X. Chen, Linear approximation of shock reflection at a wedge with large angle, Commun. Partial Differ. Equ., 21 (1996), 1103-1118.  doi: 10.1080/03605309608821219.  Google Scholar

[11]

S. X. Chen, Study on mach reflection and mach configuration, Proc. Sympos. Appl. Math., 67 (2009), 53-71.  doi: 10.1090/psapm/067.1/2605212.  Google Scholar

[12]

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

[13]

V. Elling and T. P. Liu, Supersonic flow onto a solid wedge, Commun. Pure Appl. Math., 61 (2008), 1347-1448.  doi: 10.1002/cpa.20231.  Google Scholar

[14]

J. Glimm and A. J. Majda, Multidimensional Hyperbolic Problems and Computations, Springer-Verlag, NewYork, 1991. doi: 10.1007/978-1-4613-9121-0_8.  Google Scholar

[15]

Q. Han, Nonlinear Elliptic Equations of the Second Order, American Mathematical Society, Providence, RI, 2016.  Google Scholar

[16]

E. Harabetian, Diffraction of a weak shock by a wedge, Commun. Pure. Appl. Math., 40 (1987), 849-863.  doi: 10.1002/cpa.3160400608.  Google Scholar

[17]

H. Hornung, Regular and mach reflection of shock waves, Annu. Rev. Fluid Mech., 18 (1986), 33-58.   Google Scholar

[18]

J. K. Hunter, Transverse diffraction of nonlinear waves and singular rays, SIAM J. Appl. Math., 48 (1988), 1-37.  doi: 10.1137/0148001.  Google Scholar

[19]

J. K. Hunter and A. M. Tesdall, Weak shock reflection, in A celebration of Mathematical Modeling, Springer, (2004), 93–112.  Google Scholar

[20]

J. B. Keller and A. Blank, Diffraction and reflection of pulses by wedges and corners, Commun. Pure Appl. Math., 4 (1951), 75-94.  doi: 10.1002/cpa.3160040109.  Google Scholar

[21]

E. H. Kim, A global subsonic solution to an interacting transonic shock for the self-similar nonlinear wave equation, J. Differ. Equ., 248 (2010), 2906-2930.  doi: 10.1016/j.jde.2010.02.021.  Google Scholar

[22]

J. Q. Li, T. Zhang and S. L. Yang, The Two-dimensional Riemann Problem in Gas Dynamics, Pitman Monographs, Vol. 98, 1998.  Google Scholar

[23]

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: 10.2307/2000717.  Google Scholar

[24]

G. M. Lieberman, Oblique Derivative Problems for Elliptic Equations, World Scientific, 2013. doi: 10.1142/8679.  Google Scholar

[25]

E. Mach, Uber den verlauf von funkenwellen in der ebene und im raume, Sitzungsbr. Akad. Wiss. Wien, 78 (1878), 819-838.   Google Scholar

[26]

C. S. Morawetz, Potential theory for regular and mach reflection of a shock at a wedge, Commun. Pure Appl. Math., 47 (1994), 593-624.  doi: 10.1002/cpa.3160470502.  Google Scholar

[27]

M. Rigby, Transonic Shock Waves and Free Boundary Problems for the Nonlinear Wave System, PhD thesis, University of Oxford, 2018. Google Scholar

[28]

D. Serre, Shock Reflection in Gas Dynamics, Handbook of mathematical fluid dynamics, Elsevier, 2007.  Google Scholar

[29]

N. S. Trudinger, On an interpolation inequality and its applications to nonlinear elliptic equations, Proc. Amer. Math. Soc., 95 (1985), 73-78.  doi: 10.2307/2045576.  Google Scholar

[30]

J. von Neumann, Oblique Reflection of Shocks, Bureau of Ordinance, Explosives Research Report, 1943. Google Scholar

[31]

E. Zeidler, Nonlinear Functional Analysis and its Applications I, Springer-Verlag, NewYork, 1986. doi: 10.1007/978-1-4612-4838-5.  Google Scholar

[32]

Y. X. Zheng, Two-dimensional regular shock reflection for the pressure gradient system of conservation laws, Acta Math. Appl. Sin. Engl. Ser., 22 (2006), 177-210.  doi: 10.1007/s10255-006-0296-5.  Google Scholar

[33]

Y. X. Zheng, Systems of Conservation Laws: Two-dimensional Riemann Problems, Birkhäuser: Boston, 2001. doi: 10.1007/978-1-4612-0141-0.  Google Scholar

Figure 1.  Supersonic regular shock reflection configuration(left); subsonic regular shock reflection configuration (right)}
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Figure 2.  Hypothetical curves
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