# American Institute of Mathematical Sciences

July & August  2021, 20(7&8): 2665-2685. doi: 10.3934/cpaa.2021048

## Radon measure solutions for steady compressible hypersonic-limit Euler flows passing cylindrically symmetric conical bodies

 1 Center for Partial Differential Equations, School of Mathematical Sciences, East China Normal University, Shanghai 200241, China 2 Department of Mathematics, Shanghai Normal University, Shanghai 200234, China 3 School of Mathematical Sciences, and Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai 200241, China

* Corresponding author

Received  November 2020 Revised  February 2021 Published  July & August 2021 Early access  March 2021

Fund Project: The research of Aifang Qu is supported by the National Natural Science Foundation of China (NNSFC) under Grants No. 11571357, No. 11871218, and No. 12071298; Hairong Yuan is supported by NNSFC under Grants No. 11871218, No. 12071298, and by the Science and Technology Commission of Shanghai Municipality (STCSM) under Grant No. 18dz2271000

We study steady uniform hypersonic-limit Euler flows passing a finite cylindrically symmetric conical body in the Euclidean space $\mathbb{R}^3$, and its interaction with downstream static gas lying behind the tail of the body. Motivated by Newton's theory of infinite-thin shock layers, we propose and construct Radon measure solutions with density containing Dirac measures supported on surfaces and prove the Newton-Busemann pressure law of hypersonic aerodynamics. It happens that if the pressure of the downstream static gas is quite large, the Radon measure solution terminates at a finite distance from the tail of the body. The main difficulty of the analysis is a correct definition of Radon measure solutions. The results are helpful to understand mathematically some physical phenomena and formulas about hypersonic inviscid flows.

Citation: Yunjuan Jin, Aifang Qu, Hairong Yuan. Radon measure solutions for steady compressible hypersonic-limit Euler flows passing cylindrically symmetric conical bodies. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2665-2685. doi: 10.3934/cpaa.2021048
##### References:
 [1] J. D. Jr. Anderson, Modern Compressible Flow: With Historical Perspective, 3$^rd$ edition, McGraw-Hill, 2003. [2] J. D. Jr. Anderson, Hypersonic and High-Temperature Gas Dynamics, 2$^nd$ edition, AIAA, 2006. [3] S. Chen, A free boundary value problem of Euler system arising in supersonic flow past a curved cone, Tohoku Math. J., 54 (2002), 105-120. [4] S. Chen, Existence of stationary supersonic flows past a pointed body, Arch. Ration. Mech. Anal., 156 (2001), 141-181.  doi: 10.1007/s002050100121. [5] S. Chen, Z. Geng and D. Li, The existence and stability of conic shock waves, J. Math. Anal. Appl., 277 (2003), 512-532.  doi: 10.1016/S0022-247X(02)00581-4. [6] S. Chen and D. Li., Supersonic flow past a symmetrically curved cone, Indiana U. Math. J., 49 (2000), 1411-1435.  doi: 10.1512/iumj.2000.49.1928. [7] S. Chen and D. Li, Conical shock waves for an isentropic Euler system, P. Roy. Soc. Edinb. A., 135 (2005), 1109-1127.  doi: 10.1017/S0308210500004297. [8] S. Chen and D. Li, Conical shock waves in supersonic flow, J. Differ. Equ., 269 (2020), 595-611.  doi: 10.1016/j.jde.2019.12.018. [9] S. Chen, Z. Xin and H. Yin, Global shock waves for the supersonic flow past a perturbed cone, Commun. Math. Phys., 228 (2002), 47-84.  doi: 10.1007/s002200200652. [10] S. Chen and C. Yi, Global solutions for supersonic flow past a delta wing, SIAM J. Math. Anal., 47 (2015), 80-126.  doi: 10.1137/140963157. [11] G. Q. Chen and H. Liu, Formation of $\delta$-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids, SIAM J. Math. Anal., 34 (2003), 925-938.  doi: 10.1137/S0036141001399350. [12] G. Q. Chen and H. Liu, Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids, Physica D. Nonlinear Phenomena, 189 (2004), 141-165.  doi: 10.1016/j.physd.2003.09.039. [13] H. Cheng and H. Yang, Delta shock waves as limits of vanishing viscosity for 2-D steady pressureless isentropic flow, Acta Appl. Math., 113 (2011), 323-348.  doi: 10.1007/s10440-010-9602-6. [14] G. G. Chernyi, Introduction to Hypersonic Flow, Academic Press, New York and London, 1961. [15] D. Cui and H. Yin, Global supersonic conic shock wave for the steady supersonic flow past a cone: polytropic gas, J. Differ. Equ., 246 (2009), 641-669.  doi: 10.1016/j.jde.2008.07.031. [16] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 2015. [17] D. Hu and Y. Zhang, Global conical shock wave for the steady supersonic flow past a curved cone, SIAM J. Math. Anal., 51 (2019), 2372-2389.  doi: 10.1137/18M1179924. [18] F. Huang and Z. Wang, Well posedness for pressureless flow, Commun. Math. Phys., 222 (2001), 117-146.  doi: 10.1007/s002200100506. [19] Y. Jin, A. Qu and H. Yuan, On two-dimensional steady Hypersonic-limit Euler flows passing ramps and Radon measure solutions of compressible Euler equations, preprint, arXiv: 1909.03624v1. [20] J. Kuang, W. Xiang and Y. Zhang, Hypersonic similarity for the two dimensional steady potential flow with large data, Annales de l'Institut Henri Poincaré C, Analyse Non Linéaire, 37 (2020), 1379-1423.  doi: 10.1016/j.anihpc.2020.05.002. [21] K. Louie and J. R. Ockendon, Mathematical aspects of the theory of inviscid hypersonic flow, Philosophical Transactions of the Royal Society of London. Series A., 335 (1991), 121-138.  doi: 10.1098/rsta.1991.0039. [22] M. Nedeljkov, Shadow waves: entropies and interactions for delta and singular shocks, Arch. Ration. Mech. Anal., 197 (2010), 489-537.  doi: 10.1007/s00205-009-0281-2. [23] A. Paiva, Formation of $\delta$-shock waves in isentropic fluids, Zeitschrift Für Angewandte Mathematik und Physik, 71 (2020), 110. doi: 10.1007/s00033-020-01332-6. [24] A. Qu, L. Wang and H. Yuan, Radon measure solutions for steady hypersonic-limit Euler flows passing two-dimensional finite non-symmetric obstacles and interactions of free concentration layers, Commun. Math. Sci., to appear. [25] A. Qu and H. Yuan, Radon measure solutions for steady compressible Euler equations of hypersonic-limit conical flows and Newton's sine-squared law, J. Differ. Equ., 269 (2020), 495-522.  doi: 10.1016/j.jde.2019.12.012. [26] A. Qu, H. Yuan and Q. Zhao, Hypersonic limit of two-dimensional steady compressible Euler flows passing a straight wedge, Zeitschrift Für Angewandte Mathematik und Mechanik, 100 (2020), e201800225. doi: 10.1002/zamm. 201800225. [27] Z. Wang and Y. Zhang, Steady supersonic flow past a curved cone, J. Differ. Equ., 247 (2009), 1817-1850.  doi: 10.1016/j.jde.2009.05.010. [28] Z. Wang and Q. Zhang, The Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations, Acta Math. Sci., 32 (2012), 825-841.  doi: 10.1016/S0252-9602(12)60064-2.

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##### References:
 [1] J. D. Jr. Anderson, Modern Compressible Flow: With Historical Perspective, 3$^rd$ edition, McGraw-Hill, 2003. [2] J. D. Jr. Anderson, Hypersonic and High-Temperature Gas Dynamics, 2$^nd$ edition, AIAA, 2006. [3] S. Chen, A free boundary value problem of Euler system arising in supersonic flow past a curved cone, Tohoku Math. J., 54 (2002), 105-120. [4] S. Chen, Existence of stationary supersonic flows past a pointed body, Arch. Ration. Mech. Anal., 156 (2001), 141-181.  doi: 10.1007/s002050100121. [5] S. Chen, Z. Geng and D. Li, The existence and stability of conic shock waves, J. Math. Anal. Appl., 277 (2003), 512-532.  doi: 10.1016/S0022-247X(02)00581-4. [6] S. Chen and D. Li., Supersonic flow past a symmetrically curved cone, Indiana U. Math. J., 49 (2000), 1411-1435.  doi: 10.1512/iumj.2000.49.1928. [7] S. Chen and D. Li, Conical shock waves for an isentropic Euler system, P. Roy. Soc. Edinb. A., 135 (2005), 1109-1127.  doi: 10.1017/S0308210500004297. [8] S. Chen and D. Li, Conical shock waves in supersonic flow, J. Differ. Equ., 269 (2020), 595-611.  doi: 10.1016/j.jde.2019.12.018. [9] S. Chen, Z. Xin and H. Yin, Global shock waves for the supersonic flow past a perturbed cone, Commun. Math. Phys., 228 (2002), 47-84.  doi: 10.1007/s002200200652. [10] S. Chen and C. Yi, Global solutions for supersonic flow past a delta wing, SIAM J. Math. Anal., 47 (2015), 80-126.  doi: 10.1137/140963157. [11] G. Q. Chen and H. Liu, Formation of $\delta$-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids, SIAM J. Math. Anal., 34 (2003), 925-938.  doi: 10.1137/S0036141001399350. [12] G. Q. Chen and H. Liu, Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids, Physica D. Nonlinear Phenomena, 189 (2004), 141-165.  doi: 10.1016/j.physd.2003.09.039. [13] H. Cheng and H. Yang, Delta shock waves as limits of vanishing viscosity for 2-D steady pressureless isentropic flow, Acta Appl. Math., 113 (2011), 323-348.  doi: 10.1007/s10440-010-9602-6. [14] G. G. Chernyi, Introduction to Hypersonic Flow, Academic Press, New York and London, 1961. [15] D. Cui and H. Yin, Global supersonic conic shock wave for the steady supersonic flow past a cone: polytropic gas, J. Differ. Equ., 246 (2009), 641-669.  doi: 10.1016/j.jde.2008.07.031. [16] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 2015. [17] D. Hu and Y. Zhang, Global conical shock wave for the steady supersonic flow past a curved cone, SIAM J. Math. Anal., 51 (2019), 2372-2389.  doi: 10.1137/18M1179924. [18] F. Huang and Z. Wang, Well posedness for pressureless flow, Commun. Math. Phys., 222 (2001), 117-146.  doi: 10.1007/s002200100506. [19] Y. Jin, A. Qu and H. Yuan, On two-dimensional steady Hypersonic-limit Euler flows passing ramps and Radon measure solutions of compressible Euler equations, preprint, arXiv: 1909.03624v1. [20] J. Kuang, W. Xiang and Y. Zhang, Hypersonic similarity for the two dimensional steady potential flow with large data, Annales de l'Institut Henri Poincaré C, Analyse Non Linéaire, 37 (2020), 1379-1423.  doi: 10.1016/j.anihpc.2020.05.002. [21] K. Louie and J. R. Ockendon, Mathematical aspects of the theory of inviscid hypersonic flow, Philosophical Transactions of the Royal Society of London. Series A., 335 (1991), 121-138.  doi: 10.1098/rsta.1991.0039. [22] M. Nedeljkov, Shadow waves: entropies and interactions for delta and singular shocks, Arch. Ration. Mech. Anal., 197 (2010), 489-537.  doi: 10.1007/s00205-009-0281-2. [23] A. Paiva, Formation of $\delta$-shock waves in isentropic fluids, Zeitschrift Für Angewandte Mathematik und Physik, 71 (2020), 110. doi: 10.1007/s00033-020-01332-6. [24] A. Qu, L. Wang and H. Yuan, Radon measure solutions for steady hypersonic-limit Euler flows passing two-dimensional finite non-symmetric obstacles and interactions of free concentration layers, Commun. Math. Sci., to appear. [25] A. Qu and H. Yuan, Radon measure solutions for steady compressible Euler equations of hypersonic-limit conical flows and Newton's sine-squared law, J. Differ. Equ., 269 (2020), 495-522.  doi: 10.1016/j.jde.2019.12.012. [26] A. Qu, H. Yuan and Q. Zhao, Hypersonic limit of two-dimensional steady compressible Euler flows passing a straight wedge, Zeitschrift Für Angewandte Mathematik und Mechanik, 100 (2020), e201800225. doi: 10.1002/zamm. 201800225. [27] Z. Wang and Y. Zhang, Steady supersonic flow past a curved cone, J. Differ. Equ., 247 (2009), 1817-1850.  doi: 10.1016/j.jde.2009.05.010. [28] Z. Wang and Q. Zhang, The Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations, Acta Math. Sci., 32 (2012), 825-841.  doi: 10.1016/S0252-9602(12)60064-2.
The upstream hypersonic-limit flow is separated from the downstream static gas by an axially-symmetric free concentration interface
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