Global solutions of shock reflection problem for the pressure gradient system

Hanchun Yang; Meimei Zhang; Qin Wang

doi:10.3934/cpaa.2020150

Article Contents

2020, Volume 19, Issue 6: 3387-3428. Doi: 10.3934/cpaa.2020150

This issue Previous Article The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities Next Article Existence results for quasilinear Schrödinger equations with a general nonlinearity

Global solutions of shock reflection problem for the pressure gradient system

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

^* Corresponding author
^* Corresponding author

Received: December 2019

Revised: December 2019

Published: March 2020

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))

Abstract Full Text(HTML) Figure(2) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 35L50, 35L67, 76H05; Secondary: 35J67.

Citation:

Full Text(HTML)

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

Download: Full-size image PowerPoint slide

Figure 2. Hypothetical curves

Download: Full-size image PowerPoint slide

Related Papers

Cited by

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.