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Global solutions of shock reflection problem for the pressure gradient system
Department of Mathematics and Statistics, Yunnan University, Kunming, 650091, China |
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.
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] |
G. Ben-Dor, Shock Wave Reflection Phenomena, Springer-Verlag, NewYork, 2007. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
![]() ![]() |
[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. |
[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. |
[12] |
R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Springer-Verlag, NewYork, 1948. |
[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. |
[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. |
[15] |
Q. Han, Nonlinear Elliptic Equations of the Second Order, American Mathematical Society, Providence, RI, 2016. |
[16] |
E. Harabetian,
Diffraction of a weak shock by a wedge, Commun. Pure. Appl. Math., 40 (1987), 849-863.
doi: 10.1002/cpa.3160400608. |
[17] |
H. Hornung,
Regular and mach reflection of shock waves, Annu. Rev. Fluid Mech., 18 (1986), 33-58.
|
[18] |
J. K. Hunter,
Transverse diffraction of nonlinear waves and singular rays, SIAM J. Appl. Math., 48 (1988), 1-37.
doi: 10.1137/0148001. |
[19] |
J. K. Hunter and A. M. Tesdall, Weak shock reflection, in A celebration of Mathematical Modeling, Springer, (2004), 93–112. |
[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. |
[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. |
[22] |
J. Q. Li, T. Zhang and S. L. Yang, The Two-dimensional Riemann Problem in Gas Dynamics, Pitman Monographs, Vol. 98, 1998. |
[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. |
[24] |
G. M. Lieberman, Oblique Derivative Problems for Elliptic Equations, World Scientific, 2013.
doi: 10.1142/8679. |
[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] |
M. Rigby, Transonic Shock Waves and Free Boundary Problems for the Nonlinear Wave System, PhD thesis, University of Oxford, 2018. |
[28] |
D. Serre, Shock Reflection in Gas Dynamics, Handbook of mathematical fluid dynamics, Elsevier, 2007. |
[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. |
[30] |
J. von Neumann, Oblique Reflection of Shocks, Bureau of Ordinance, Explosives Research Report, 1943. |
[31] |
E. Zeidler, Nonlinear Functional Analysis and its Applications I, Springer-Verlag, NewYork, 1986.
doi: 10.1007/978-1-4612-4838-5. |
[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. |
[33] |
Y. X. Zheng, Systems of Conservation Laws: Two-dimensional Riemann Problems, Birkhäuser: Boston, 2001.
doi: 10.1007/978-1-4612-0141-0. |
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] |
G. Ben-Dor, Shock Wave Reflection Phenomena, Springer-Verlag, NewYork, 2007. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
![]() ![]() |
[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. |
[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. |
[12] |
R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Springer-Verlag, NewYork, 1948. |
[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. |
[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. |
[15] |
Q. Han, Nonlinear Elliptic Equations of the Second Order, American Mathematical Society, Providence, RI, 2016. |
[16] |
E. Harabetian,
Diffraction of a weak shock by a wedge, Commun. Pure. Appl. Math., 40 (1987), 849-863.
doi: 10.1002/cpa.3160400608. |
[17] |
H. Hornung,
Regular and mach reflection of shock waves, Annu. Rev. Fluid Mech., 18 (1986), 33-58.
|
[18] |
J. K. Hunter,
Transverse diffraction of nonlinear waves and singular rays, SIAM J. Appl. Math., 48 (1988), 1-37.
doi: 10.1137/0148001. |
[19] |
J. K. Hunter and A. M. Tesdall, Weak shock reflection, in A celebration of Mathematical Modeling, Springer, (2004), 93–112. |
[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. |
[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. |
[22] |
J. Q. Li, T. Zhang and S. L. Yang, The Two-dimensional Riemann Problem in Gas Dynamics, Pitman Monographs, Vol. 98, 1998. |
[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. |
[24] |
G. M. Lieberman, Oblique Derivative Problems for Elliptic Equations, World Scientific, 2013.
doi: 10.1142/8679. |
[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] |
M. Rigby, Transonic Shock Waves and Free Boundary Problems for the Nonlinear Wave System, PhD thesis, University of Oxford, 2018. |
[28] |
D. Serre, Shock Reflection in Gas Dynamics, Handbook of mathematical fluid dynamics, Elsevier, 2007. |
[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. |
[30] |
J. von Neumann, Oblique Reflection of Shocks, Bureau of Ordinance, Explosives Research Report, 1943. |
[31] |
E. Zeidler, Nonlinear Functional Analysis and its Applications I, Springer-Verlag, NewYork, 1986.
doi: 10.1007/978-1-4612-4838-5. |
[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. |
[33] |
Y. X. Zheng, Systems of Conservation Laws: Two-dimensional Riemann Problems, Birkhäuser: Boston, 2001.
doi: 10.1007/978-1-4612-0141-0. |


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