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doi: 10.3934/cpaa.2021014

## Global solutions of a two-dimensional Riemann problem for the pressure gradient system

 1 Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK, School of Mathematical Sciences, Fudan University, Shanghai 200433, China, AMSS, Chinese Academy of Sciences, Beijing 100190, China 2 Department of Mathematics, Yunnan University, Kunming 650091, China 3 School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China, Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK

* Corresponding author

Dedicated to Professor Shuxing Chen on the occasion of his 80th birthday

Received  August 2020 Revised  November 2020 Published  February 2021

Fund Project: Gui-Qiang G. Chen's research was supported in part by the UK Engineering and Physical Sciences Research Council under Grant EP/L015811/1 and the Royal Society–Wolfson Research Merit Award WM090014 (UK). Qin Wang's research was supported in part by National Natural Science Foundation of China (11761077), China Scholarship Council (201807035046), and the Key Project of Yunnan Provincial Science and Technology Department and Yunnan University (No.2018FY001-014). Shengguo Zhu's research was supported in part by the Royal Society–Newton International Fellowships NF170015 and the Monash University–Robert Bartnik Visiting Fellowship. Qin Wang would also like to thank the hospitality and support of the Mathematical Institute, University of Oxford, during his visit in 2019–20

We are concerned with a two-dimensional Riemann problem for the pressure gradient system that is a hyperbolic system of conservation laws. The Riemann initial data consist of four constant states in four sectorial regions such that two shocks and two vortex sheets are generated between the adjacent states. The solutions keep the four constant states and four planar waves outside the outer sonic circle in the self-similar coordinates, while the two shocks keep planar until meeting the outer sonic circle at two different points and then generate a diffracted shock to connect these points, whose location is apriori unknown. Then the problem can be formulated as a free boundary problem, in which the diffracted transonic shock is the one-phase free boundary to connect the two points, while the other part of the sonic circle forms a fixed boundary. We establish the global existence of a solution and the optimal Lipschitz regularity of both the diffracted shock across the two points and the solution across the outer sonic boundary. Then this Riemann problem is solved globally, whose solution contains two vortex sheets and one global shock containing the two originally separated shocks generated by the Riemann data.

Citation: Gui-Qiang G. Chen, Qin Wang, Shengguo Zhu. Global solutions of a two-dimensional Riemann problem for the pressure gradient system. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021014
##### References:
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Chen, Construction of solutions to M-D Riemann problems for a $2\times 2$ quasilinear hyperbolic system, Chinese Ann. Math., 18B (1997), 345-358.   Google Scholar [16] S. X. Chen and B. Fang, Stability of transonic shocks in supersonic flow past a wedge, J. Differ. Equ., 233 (2007), 105-135.  doi: 10.1016/j.jde.2006.09.020.  Google Scholar [17] S. X. Chen and A. Qu, Two-dimensional Riemann problems for Chaplygin gas, SIAM J. Math. Anal., 44 (2012), 2146-2178.  doi: 10.1137/110838091.  Google Scholar [18] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, Berlin, 2016. doi: 10.1007/978-3-662-49451-6.  Google Scholar [19] J. V. Egorov and V. A. Kondrat'ev, The oblique derivative problem, Mathematics of the USSR-Sbornik, 7 (1969), 139 pp.  Google Scholar [20] 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.  Google Scholar [21] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.  Google Scholar [22] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Commun. Pure Appl. Math., 18 (1965), 697-715.  doi: 10.1002/cpa.3160180408.  Google Scholar [23] L. Hörmander, Pseudo-differential operators and non-elliptic boundary problems, Ann. Math., 83 (1966), 129-209.  doi: 10.2307/1970473.  Google Scholar [24] B. L. Keyfitz and S. Canic, Riemann problems for the two-dimensional unsteady transonic small disturbance equation, SIAM J. Appl. Math., 58 (1998), 636-665.  doi: 10.1137/S0036139996300.  Google Scholar [25] 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 [26] P. Lax, Hyperbolic systems of conservation laws Ⅱ, Commun. Pure Appl. Math., 4 (1957), 537-566.  doi: 10.1002/cpa.3160100406.  Google Scholar [27] J. Li, T. Zhang, and S. Yang, The Two-Dimensional Riemann Problem in Gas Dynamics, Monographs and Surveys in Pure and Applied Mathematics, Vol. 98, Chapman & Hall/CRC, Longman, Harlow, 1998.  Google Scholar [28] Y. F. Li and Y. M. Cao, Large-particle difference method with second-order accuracy in gasdynamics, Sci. China, 28A (1985), 1024-1035.   Google Scholar [29] 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 [30] 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 [31] P. R. Popivanov and D. K. Palagachev, The Degenerate Oblique Derivative Problem for Elliptic and Parabolic Equations, Akademie Verlag, Berlin, 1997.  Google Scholar [32] B. Riemann, Über die Fortpflanzung ebener Luftvellen von endlicher Schwingungsweite, Gött. Abh. Math. Cl., 8 (1860), 43-65.   Google Scholar [33] J. Smoller., The Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar [34] B. Winzell, A boundary value problem with an oblique derivative, Commun. Partial Differ. Equ., 6 (1981), 305-328.  doi: 10.1080/03605308108820179.  Google Scholar [35] H. Yuan, On transonic shocks in two-dimensional variable-area ducts for steady Euler system, SIAM J. Math. Anal., 38 (2006), 1343-1370.  doi: 10.1137/050642447.  Google Scholar [36] P. Zhang, J. Li and T. Zhang, On two-dimensional Riemann problem for pressure-gradient equations of the Euler system, Discret. Contin. Dyn. Syst., 4 (1998), 609-634.  doi: 10.3934/dcds.1998.4.609.  Google Scholar [37] Y. Zheng, Existence of solutions to the transonic pressure gradient equations of the compressible Euler equations in elliptic regions, Commun. Partial Differ. Equ., 22 (1997), 1849-1868.  doi: 10.1080/03605309708821323.  Google Scholar [38] Y. Zheng, A global solution to a two-dimensional Riemann problem involving shocks as free boundaries, Acta Math. Appl. Sin., 19 (2003), 559-572.  doi: 10.1007/210255-003-0131-1.  Google Scholar [39] Y. Zheng, Two-dimensional regular shock reflection for the pressure gradient system of conservation laws, Acta Math. Appl. Sin., 22 (2006), 177-210.  doi: 10.1007/s10255-006-0296-5.  Google Scholar [40] Y. Zheng, Systems of Conservation Laws: Two-Dimensional Riemann Problems, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4612-0141-0.  Google Scholar

show all references

##### References:
 [1] R. Agarwal and D. Halt, A modified CUSP scheme in wave/particle split form for unstructured grid Euler flows, In: Frontiers of Computational Fluid Dynamics, pp.155–163, 1994. Google Scholar [2] 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 [3] M. Bae, G. Q. Chen and M. Feldman, Prandtl-Meyer Reflection Configurations, Transonic Shocks, and Free Boundary Problems, Memoirs Amer. Math. Soc. (to appear); arXiv: 1901.05916, 2020. Google Scholar [4] S. Canic, B. L. Keyfitz and E. H. Kim, Free boundary problems for the unsteady transonic small disturbance equation: Transonic regular reflection, Methods Appl. Anal., 7 (2000), 313-336.  doi: 10.4310/MAA.2000.v7.n2.a4.  Google Scholar [5] 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 [6] 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 [7] T. Chang, G. Q. Chen and S. L. Yang, On the $2$-D Riemann problem for the compressible Euler equations. I. Interaction of shocks and rarefaction waves, Discrete Contin. Dynam. Systems, 1 (1995), 555-584.  doi: 10.3934/dcds.1995.1.555.  Google Scholar [8] T. Chang, G. Q. Chen and S. L. Yang, On the $2$-D Riemann problem for the compressible Euler equations., Ⅱ. Interaction of contact discontinuities doi: 10.3934/dcds.2000.6.419.  Google Scholar [9] T. Chang and L. Hsiao, The Riemann Problem and Interaction of Waves in Gas Dynamics, Longman Scientific & Technical: Harlow; John Wiley & Sons, Inc., New York, 1989.  Google Scholar [10] G. Q. Chen, X. 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 [11] 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.  Google Scholar [12] G. Q. Chen and M. Feldman, The Mathematics of Shock Reflection-Diffraction and Von Neumann's Conjectures, Princeton University Press, Princeton, 2018.   Google Scholar [13] G. Q. Chen and P. LeFloch, Entropy flux-splittings for hyperbolic conservation laws, Commun. Pure Appl. Math., 48 (1995), 691-729.  doi: 10.1002/cpa.3160480703.  Google Scholar [14] S. X. Chen, Multidimensional Riemann problem for semilinear wave equations, Commun. Partial Differ. Equ., 17 (1992), 715-736.  doi: 10.1080/03605309208820861.  Google Scholar [15] S. X. Chen, Construction of solutions to M-D Riemann problems for a $2\times 2$ quasilinear hyperbolic system, Chinese Ann. Math., 18B (1997), 345-358.   Google Scholar [16] S. X. Chen and B. Fang, Stability of transonic shocks in supersonic flow past a wedge, J. Differ. Equ., 233 (2007), 105-135.  doi: 10.1016/j.jde.2006.09.020.  Google Scholar [17] S. X. Chen and A. Qu, Two-dimensional Riemann problems for Chaplygin gas, SIAM J. Math. Anal., 44 (2012), 2146-2178.  doi: 10.1137/110838091.  Google Scholar [18] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, Berlin, 2016. doi: 10.1007/978-3-662-49451-6.  Google Scholar [19] J. V. Egorov and V. A. Kondrat'ev, The oblique derivative problem, Mathematics of the USSR-Sbornik, 7 (1969), 139 pp.  Google Scholar [20] 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.  Google Scholar [21] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.  Google Scholar [22] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Commun. Pure Appl. Math., 18 (1965), 697-715.  doi: 10.1002/cpa.3160180408.  Google Scholar [23] L. Hörmander, Pseudo-differential operators and non-elliptic boundary problems, Ann. Math., 83 (1966), 129-209.  doi: 10.2307/1970473.  Google Scholar [24] B. L. Keyfitz and S. Canic, Riemann problems for the two-dimensional unsteady transonic small disturbance equation, SIAM J. Appl. Math., 58 (1998), 636-665.  doi: 10.1137/S0036139996300.  Google Scholar [25] 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 [26] P. Lax, Hyperbolic systems of conservation laws Ⅱ, Commun. Pure Appl. Math., 4 (1957), 537-566.  doi: 10.1002/cpa.3160100406.  Google Scholar [27] J. Li, T. Zhang, and S. Yang, The Two-Dimensional Riemann Problem in Gas Dynamics, Monographs and Surveys in Pure and Applied Mathematics, Vol. 98, Chapman & Hall/CRC, Longman, Harlow, 1998.  Google Scholar [28] Y. F. Li and Y. M. Cao, Large-particle difference method with second-order accuracy in gasdynamics, Sci. China, 28A (1985), 1024-1035.   Google Scholar [29] 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 [30] 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 [31] P. R. Popivanov and D. K. Palagachev, The Degenerate Oblique Derivative Problem for Elliptic and Parabolic Equations, Akademie Verlag, Berlin, 1997.  Google Scholar [32] B. Riemann, Über die Fortpflanzung ebener Luftvellen von endlicher Schwingungsweite, Gött. Abh. Math. Cl., 8 (1860), 43-65.   Google Scholar [33] J. Smoller., The Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar [34] B. Winzell, A boundary value problem with an oblique derivative, Commun. Partial Differ. Equ., 6 (1981), 305-328.  doi: 10.1080/03605308108820179.  Google Scholar [35] H. Yuan, On transonic shocks in two-dimensional variable-area ducts for steady Euler system, SIAM J. Math. Anal., 38 (2006), 1343-1370.  doi: 10.1137/050642447.  Google Scholar [36] P. Zhang, J. Li and T. Zhang, On two-dimensional Riemann problem for pressure-gradient equations of the Euler system, Discret. Contin. Dyn. Syst., 4 (1998), 609-634.  doi: 10.3934/dcds.1998.4.609.  Google Scholar [37] Y. Zheng, Existence of solutions to the transonic pressure gradient equations of the compressible Euler equations in elliptic regions, Commun. Partial Differ. Equ., 22 (1997), 1849-1868.  doi: 10.1080/03605309708821323.  Google Scholar [38] Y. Zheng, A global solution to a two-dimensional Riemann problem involving shocks as free boundaries, Acta Math. Appl. Sin., 19 (2003), 559-572.  doi: 10.1007/210255-003-0131-1.  Google Scholar [39] Y. Zheng, Two-dimensional regular shock reflection for the pressure gradient system of conservation laws, Acta Math. Appl. Sin., 22 (2006), 177-210.  doi: 10.1007/s10255-006-0296-5.  Google Scholar [40] Y. Zheng, Systems of Conservation Laws: Two-Dimensional Riemann Problems, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4612-0141-0.  Google Scholar
The general Riemann initial data
The configuration of the four initial waves
The Riemann data and the global solution when $\alpha_1 = 0$
Hypothetical curves
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