# American Institute of Mathematical Sciences

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

 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.

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 and 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. [2] M. Bae, G. Q. Chen and M. Feldman, Prandtl-meyer reflection configurations, transonic shocks, and free boundary problems, preprint, arXiv: 1901.05916. [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. [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. [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. [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. [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. [8] G. Q. Chen and M. Feldman, The Mathematics of Shock Reflection-diffraction and Von Neumann's Conjectures, 359, Princeton University Press, 2018. [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. [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. [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. [12] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, 21, Springer Science & Business Media, 1999. [13] C. Fletcher and W. Bleakney, The Mach reflection of shock waves at nearly glancing incidence, Rev. Mod. Phys., 23 (1951), 271. [14] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2015. [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. [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. [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. [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. [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. [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. [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. [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. [23] M. J. Lighthill, The diffraction of blast (i), Proc. R. Soc. A, 198 (1949), 454-470.  doi: 10.1098/rspa.1949.0113. [24] M. J. Lighthill, The diffraction of blast (ii), Proc. R. Soc. A, 200 (1950), 554-565.  doi: 10.1098/rspa.1950.0037. [25] E. Mach, Uber den verlauf von funkenwellen in der ebene und im raume, Sitzungsbr. Akad. Wiss. Wien, 78 (1878), 819-838. [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. [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. [28] J. Von Neumann and A. Taub, Collected Works, Vol. 1-6, Theory of Games, Astrophysics, Hydrodynamics and Meteorology, 1963. [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. [30] Y. X. Zheng, Systems of Conservation Laws: Two-dimensional Riemann Problems, 38, Springer, 2001. doi: 10.1007/978-1-4612-0141-0. [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.

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##### 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. [2] M. Bae, G. Q. Chen and M. Feldman, Prandtl-meyer reflection configurations, transonic shocks, and free boundary problems, preprint, arXiv: 1901.05916. [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. [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. [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. [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. [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. [8] G. Q. Chen and M. Feldman, The Mathematics of Shock Reflection-diffraction and Von Neumann's Conjectures, 359, Princeton University Press, 2018. [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. [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. [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. [12] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, 21, Springer Science & Business Media, 1999. [13] C. Fletcher and W. Bleakney, The Mach reflection of shock waves at nearly glancing incidence, Rev. Mod. Phys., 23 (1951), 271. [14] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2015. [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. [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. [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. [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. [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. [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. [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. [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. [23] M. J. Lighthill, The diffraction of blast (i), Proc. R. Soc. A, 198 (1949), 454-470.  doi: 10.1098/rspa.1949.0113. [24] M. J. Lighthill, The diffraction of blast (ii), Proc. R. Soc. A, 200 (1950), 554-565.  doi: 10.1098/rspa.1950.0037. [25] E. Mach, Uber den verlauf von funkenwellen in der ebene und im raume, Sitzungsbr. Akad. Wiss. Wien, 78 (1878), 819-838. [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. [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. [28] J. Von Neumann and A. Taub, Collected Works, Vol. 1-6, Theory of Games, Astrophysics, Hydrodynamics and Meteorology, 1963. [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. [30] Y. X. Zheng, Systems of Conservation Laws: Two-dimensional Riemann Problems, 38, Springer, 2001. doi: 10.1007/978-1-4612-0141-0. [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.
Shock $S_0$ passes the wedge at $t = 0$
Shock diffraction configuration
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