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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 |
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.
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. |
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. |
[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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