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October  2020, 19(10): 4899-4920. doi: 10.3934/cpaa.2020217

Subsonic solutions to a shock diffraction problem by a convex cornered wedge for the pressure gradient system

Yinzheng Sun 1, , Qin Wang 1, and  Kyungwoo Song 2,,

1. 

Department of Mathematics, Yunnan University, Kunming 650091, China
2. 

Department of Mathematics, Kyung Hee University, Seoul 02447, Korea

* Corresponding author

Received  December 2019 Revised  June 2020 Published  July 2020

Fund Project: The research of Qin Wang is supported by NNSF of China (No. 11761077), Project of Yunnan University (No. 2019FY003007) and Program for Innovative Research Team (in Science and Technology) in Universities of Yunnan Province. The research of Kyungwoo Song is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2019R1F1A1057766)

We establish the global existence of subsonic solutions to a two dimensional Riemann problem governed by a self-similar pressure gradient system for shock diffraction by a convex cornered wedge. Since the boundary of the subsonic region consists of a transonic shock and a part of a sonic circle, the governing equation becomes a free boundary problem for nonlinear degenerate elliptic equation of second order with a degenerate oblique derivative boundary condition. We also obtain the optimal $ C^{0,1} $-regularity of the solutions across the degenerate sonic boundary.

Keywords: Pressure gradient system, free boundary problem, shock diffraction, degenerate elliptic equation, transonic shock, Riemann problem.
Mathematics Subject Classification: Primary: 35L50, 35L67, 35J70.
Citation: Yinzheng Sun, Qin Wang, Kyungwoo Song. Subsonic solutions to a shock diffraction problem by a convex cornered wedge for the pressure gradient system. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4899-4920. doi: 10.3934/cpaa.2020217
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]

M. Bae, G. Q. Chen and M. Feldman, Prandtl-meyer reflection configurations, transonic shocks, and free boundary problems, preprint, arXiv: 1901.05916. Google Scholar

[3]

S. Canic, 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

[4]

S. Canic, B. L. Keyfitz and E. H. Kim, Free boundary problems for nonlinear wave systems: Mach stems for interacting shocks, SIAM J. Math. Anal., 37 (2006), 1947-1977.  doi: 10.1137/S003614100342989X.  Google Scholar

[5]

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

[6]

G. Q. Chen and W. Xiang, Existence and stability of global solutions of shock diffraction by wedges for potential flow, in Hyperbolic Conservation Laws and Related Analysis With Applications, (2014), 113–142. doi: 10.1007/978-3-642-39007-4_6.  Google Scholar

[7]

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

[8] G. Q. Chen and M. Feldman, The Mathematics of Shock Reflection-diffraction and Von Neumann's Conjectures, 359, Princeton University Press, 2018.   Google Scholar
[9]

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

[10]

S. X. Chen and A. F. Qu, Riemann boundary value problems and reflection of shock for the chaplygin gas, Sci. China Math., 55 (2012), 671-685.  doi: 10.1007/s11425-012-4393-z.  Google Scholar

[11]

S. X. Chen and A. F. Qu, Piston problems of two-dimensional Chaplygin gas, Chin. Ann. Math. Ser. B, 40 (2019), 843-868.  doi: 10.1007/s11401-019-0164-2.  Google Scholar

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R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, 21, Springer Science & Business Media, 1999.  Google Scholar

[13]

C. Fletcher and W. Bleakney, The Mach reflection of shock waves at nearly glancing incidence, Rev. Mod. Phys., 23 (1951), 271.  Google Scholar

[14]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2015.  Google Scholar

[15]

J. Glimm and A. J. Majda, Multidimensional Hyperbolic Problems and Computations, 29, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4613-9121-0.  Google Scholar

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J. K. Hunter and J. B. Keller, Weak shock diffraction, Wave Motion, 6 (1984), 79-89.  doi: 10.1016/0165-2125(84)90024-6.  Google Scholar

[17]

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

[18]

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

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G. M. Lieberman, The perron process applied to oblique derivative problems, Adv. Math., 55 (1985), 161-172.  doi: 10.1016/0001-8708(85)90019-2.  Google Scholar

[20]

G. M. Lieberman, Mixed boundary value problems for elliptic and parabolic differential equations of second order, J. Math. Anal. Appl., 113 (1986), 422-440.  doi: 10.1016/0022-247X(86)90314-8.  Google Scholar

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G. M. Lieberman, Oblique derivative problems in lipschitz domains (ii): discontinuous boundary data, J. Reine Angew. Math., 389 (1988), 1-21.  doi: 10.1515/crll.1988.389.1.  Google Scholar

[22]

G. M. Lieberman, Optimal hölder regularity for mixed boundary value problems, J. Math. Anal. Appl., 143 (1989), 572-586.  doi: 10.1016/0022-247X(89)90061-9.  Google Scholar

[23]

M. J. Lighthill, The diffraction of blast (i), Proc. R. Soc. A, 198 (1949), 454-470.  doi: 10.1098/rspa.1949.0113.  Google Scholar

[24]

M. J. Lighthill, The diffraction of blast (ii), Proc. R. Soc. A, 200 (1950), 554-565.  doi: 10.1098/rspa.1950.0037.  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]

D. Serre, Multidimensional shock interaction for a chaplygin gas, Arch. Ration. Mech. Anal., 191 (2009), 539-577.  doi: 10.1007/s00205-008-0110-z.  Google Scholar

[28]

J. Von Neumann and A. Taub, Collected Works, Vol. 1-6, Theory of Games, Astrophysics, Hydrodynamics and Meteorology, 1963.  Google Scholar

[29]

Q. Wang, J. Q. Zhang and H. C. Yang, Two dimensional Riemann-type problem and shock diffraction for the chaplygin gas, Appl. Math. Lett., (2020), Art. 106046. doi: 10.1016/j.aml.2019.106046.  Google Scholar

[30]

Y. X. Zheng, Systems of Conservation Laws: Two-dimensional Riemann Problems, 38, Springer, 2001. doi: 10.1007/978-1-4612-0141-0.  Google Scholar

[31]

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

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]

M. Bae, G. Q. Chen and M. Feldman, Prandtl-meyer reflection configurations, transonic shocks, and free boundary problems, preprint, arXiv: 1901.05916. Google Scholar

[3]

S. Canic, 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

[4]

S. Canic, B. L. Keyfitz and E. H. Kim, Free boundary problems for nonlinear wave systems: Mach stems for interacting shocks, SIAM J. Math. Anal., 37 (2006), 1947-1977.  doi: 10.1137/S003614100342989X.  Google Scholar

[5]

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

[6]

G. Q. Chen and W. Xiang, Existence and stability of global solutions of shock diffraction by wedges for potential flow, in Hyperbolic Conservation Laws and Related Analysis With Applications, (2014), 113–142. doi: 10.1007/978-3-642-39007-4_6.  Google Scholar

[7]

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

[8] G. Q. Chen and M. Feldman, The Mathematics of Shock Reflection-diffraction and Von Neumann's Conjectures, 359, Princeton University Press, 2018.   Google Scholar
[9]

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

[10]

S. X. Chen and A. F. Qu, Riemann boundary value problems and reflection of shock for the chaplygin gas, Sci. China Math., 55 (2012), 671-685.  doi: 10.1007/s11425-012-4393-z.  Google Scholar

[11]

S. X. Chen and A. F. Qu, Piston problems of two-dimensional Chaplygin gas, Chin. Ann. Math. Ser. B, 40 (2019), 843-868.  doi: 10.1007/s11401-019-0164-2.  Google Scholar

[12]

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, 21, Springer Science & Business Media, 1999.  Google Scholar

[13]

C. Fletcher and W. Bleakney, The Mach reflection of shock waves at nearly glancing incidence, Rev. Mod. Phys., 23 (1951), 271.  Google Scholar

[14]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2015.  Google Scholar

[15]

J. Glimm and A. J. Majda, Multidimensional Hyperbolic Problems and Computations, 29, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4613-9121-0.  Google Scholar

[16]

J. K. Hunter and J. B. Keller, Weak shock diffraction, Wave Motion, 6 (1984), 79-89.  doi: 10.1016/0165-2125(84)90024-6.  Google Scholar

[17]

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

[18]

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

[19]

G. M. Lieberman, The perron process applied to oblique derivative problems, Adv. Math., 55 (1985), 161-172.  doi: 10.1016/0001-8708(85)90019-2.  Google Scholar

[20]

G. M. Lieberman, Mixed boundary value problems for elliptic and parabolic differential equations of second order, J. Math. Anal. Appl., 113 (1986), 422-440.  doi: 10.1016/0022-247X(86)90314-8.  Google Scholar

[21]

G. M. Lieberman, Oblique derivative problems in lipschitz domains (ii): discontinuous boundary data, J. Reine Angew. Math., 389 (1988), 1-21.  doi: 10.1515/crll.1988.389.1.  Google Scholar

[22]

G. M. Lieberman, Optimal hölder regularity for mixed boundary value problems, J. Math. Anal. Appl., 143 (1989), 572-586.  doi: 10.1016/0022-247X(89)90061-9.  Google Scholar

[23]

M. J. Lighthill, The diffraction of blast (i), Proc. R. Soc. A, 198 (1949), 454-470.  doi: 10.1098/rspa.1949.0113.  Google Scholar

[24]

M. J. Lighthill, The diffraction of blast (ii), Proc. R. Soc. A, 200 (1950), 554-565.  doi: 10.1098/rspa.1950.0037.  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]

D. Serre, Multidimensional shock interaction for a chaplygin gas, Arch. Ration. Mech. Anal., 191 (2009), 539-577.  doi: 10.1007/s00205-008-0110-z.  Google Scholar

[28]

J. Von Neumann and A. Taub, Collected Works, Vol. 1-6, Theory of Games, Astrophysics, Hydrodynamics and Meteorology, 1963.  Google Scholar

[29]

Q. Wang, J. Q. Zhang and H. C. Yang, Two dimensional Riemann-type problem and shock diffraction for the chaplygin gas, Appl. Math. Lett., (2020), Art. 106046. doi: 10.1016/j.aml.2019.106046.  Google Scholar

[30]

Y. X. Zheng, Systems of Conservation Laws: Two-dimensional Riemann Problems, 38, Springer, 2001. doi: 10.1007/978-1-4612-0141-0.  Google Scholar

[31]

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

Figure 1.  Shock $ S_0 $ passes the wedge at $ t = 0 $
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Figure 2.  Shock diffraction configuration
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