We are concerned with the formation of singularities and the existence of global continuous solutions of the Cauchy problem for the one-dimensional nonisentropic Euler equations for compressible fluids. For the isentropic Euler equations, we pinpoint a necessary and sufficient condition for the formation of singularities of solutions with large initial data that allow a far-field vacuum---there exists a compression in the initial data. For the nonisentropic Euler equations, we identify a sufficient condition for the formation of singularities of solutions with large initial data that allow a far-field vacuum---there exists a strong compression in the initial data. Furthermore, we identify two new phenomena---decompression and de-rarefaction---for the nonisentropic Euler flows, different from the isentropic flows, via constructing two respective solutions. For the decompression phenomenon, we construct a first global continuous nonisentropic solution, even though the initial data contain a weak compression, by solving a backward Goursat problem, so that the solution is smooth, except on several characteristic curves across which the solution has a weak discontinuity (i.e., only Lipschitz continuity). For the de-rarefaction phenomenon, we construct a continuous nonisentropic solution whose initial data contain isentropic rarefactions (i.e., without compression) and a locally stationary varying entropy profile, for which the solution still forms a shock wave in a finite time.


  1. Euler equations
  2. nonisentropic
  3. compressible flow
  4. continuous solution
  5. singularity formation
  6. far-field vacuum

MSC codes

  1. 76N15
  2. 35L65
  3. 35L67
  4. 35Q31
  5. 35A01
  6. 35B44

Get full access to this article

View all available purchase options and get full access to this article.


G. Chen, Formation of singularity and smooth wave propagation for the non-isentropic compressible Euler equations, J. Hyperbolic Differ. Equ., 8 (2011), pp. 671--690.
G. Chen, Optimal time-dependent lower bound on density for classical solutions of 1-D compressible Euler equations, Indiana Univ. Math. J., 66 (2017), pp. 725--740.
G. Chen, R. Pan, and S. Zhu, Singularity formation for the compressible Euler equations, SIAM J. Math. Anal., 49 (2017), pp. 2591--2614.
G. Chen, R. Pan, and S. Zhu, A polygonal scheme and the lower bound on density for the isentropic gas dynamics, Discrete Contin. Dyn. Syst., 39 (2019), pp. 4259--4277.
G. Chen and R. Young, Smooth solutions and singularity formation for the inhomogeneous nonlinear wave equation, J. Differential Equations, 252 (2012), pp. 2580--2595.
G. Chen and R. Young, Shock-free solutions of the compressible Euler equations, Arch. Ration. Mech. Anal., 217 (2015), pp. 1265--1293.
G. Chen, R. Young, and Q. Zhang, Shock formation in the compressible Euler equations and related systems, J. Hyperbolic Differ. Equ., 10 (2013), pp. 149--172.
G.-Q. Chen, The method of quasidecoupling for discontinuous solutions to conservation laws, Arch. Ration. Mech. Anal., 121 (1992), pp. 131--185.
S. Chen and L. Dong, Formation and construction of shock for p-system, Science in China, 44 (2001), pp. 1139--1147.
S. Chen, Z. Xin, and H. Yin, Formation and Construction of Shock Wave for Quasilinear Hyperbolic System and Its Application to Inviscid Compressible Flow, Research report, IMS, CUHK, 1999.
D. Christodoulou, The Shock Development Problem, EMS Monogr. Math., EMS, Zürich, 2019.
D. Christodoulou and S. Miao, Compressible Flow and Euler's Equations, International Press, Somerville, MA; Higher Education Press, Beijing, 2014.
R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York, 1948.
C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 4th ed., Springer-Verlag, Berlin, 2016.
F. John, Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math., 27 (1974), pp. 377--405.
D. Kong, Formation and propagation of singularities for 2 x 2 quasilinear hyperbolic systems, Trans. Amer. Math. Soc., 354 (2002), pp. 3155--3179.
P. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys., 5 (1964), pp. 611--613.
M. Lebaud, Description de la formation d'un choc dans le p-systeme, J. Math Pures Appl., 73 (1994), pp. 523--565.
T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems, Research in Applied Math. 32, Wiley-Masson, New York, 1994.
T. Li and W. Yu, Boundary Value Problems for Quasilinear Hyperbolic Systems, Duke University Press, Durham, NC, 1985.
Y. Li and S. Zhu, On regular solutions of the 3-D compressible isentropic Euler-Boltzmann equations with vacuum, Discrete Contin. Dynam. Syst., 35 (2015), pp. 3059--3086.
L. Lin, H. Liu, and T. Yang, Existence of globally bounded continuous solutions for nonisentropic gas dynamics equations, J. Math. Anal. Appl., 209 (1997), pp. 492--506.
T. Liu, Development of singularities in the nonlinear waves for quasi-linear hyperbolic partial differential equations, J. Differential Equations, 33 (1979), pp. 92--111.
J. Luk and J. Speck, Shock formation in solutions to the 2D compressible Euler equations in the presence of non-zero vorticity, Invent. Math., 214 (2018), pp. 1--169.
T. Makino, S. Ukai, and S. Kawashima, Sur la solution $\grave{{a}}$ support compact de equations d'Euler compressible, Japan J. Appl. Math., 33 (1986), pp. 249--257.
R. Pan and Y. Zhu, Singularity formation for one dimensional full Euler equations, J. Differential Equations, 261 (2016), pp. 7132--7144.
M. A. Rammaha, Formation of singularities in compressible fluids in two space dimensions, Proc. Amer. Math. Soc., 107 (1989), pp. 705--714.
T. C. Sideris, Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys., 101 (1985), pp. 475--487.
J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983.
G. G. Stokes, On a difficulty in the theory of sound, Philos. Mag. Ser. 3, 33 (1848), pp. 349--356.
B. Temple and R. Young, A paradigm for time-periodic sound wave propagation in the compressible Euler equations, Methods Appl. Anal., 16 (2009), pp. 341--364.
D. Wagner, Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions, J. Differential Equations, 68 (1987), pp. 118--136.
H. Whitney, On singularities of mappings of Euclidean space, I: Mappings of the plane into the plane, Ann. of Math., 62 (1955), pp. 374--410.

Information & Authors


Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 6280 - 6325
ISSN (online): 1095-7154


Submitted: 3 February 2020
Accepted: 9 July 2021
Published online: 4 November 2021


  1. Euler equations
  2. nonisentropic
  3. compressible flow
  4. continuous solution
  5. singularity formation
  6. far-field vacuum

MSC codes

  1. 76N15
  2. 35L65
  3. 35L67
  4. 35Q31
  5. 35A01
  6. 35B44



Funding Information

Newton International Fellowships Alumni : AL/201021
Australian Research Council https://doi.org/10.13039/501100000923 : DP170100630
National Science Foundation https://doi.org/10.13039/100000001 : DMS-1715012
Royal Society https://doi.org/10.13039/501100000288 : WM090014, NF170015
Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/E035027/1, EP/L015811/1

Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

View Options

View options


View PDF







Copy the content Link

Share with email

Email a colleague

Share on social media

On May 28, 2024, our site will enter Read Only mode for a limited time in order to complete a platform upgrade. As a result, the following functions will be temporarily unavailable: registering new user accounts, any updates to existing user accounts, access token activations, and shopping cart transactions. Contact [email protected] with any questions.