Singularity formation for one dimensional full Euler equations. (English) Zbl 1348.76125
Summary: We investigate the basic open question on the global existence v.s. finite time blow-up phenomena of classical solutions for the one-dimensional compressible Euler equations of adiabatic flow. For isentropic flows, it is well-known that the solutions develop singularity if and only if initial data contain any compression (the Riemann variables have negative spatial derivative). The situation for non-isentropic flow is not quite clear so far, due to the presence of non-constant entropy. In [G. Chen et al, “Singularity formation for compressible Euler equations”, Preprint, arXiv:1408.6775], it is shown that initial weak compressions do not necessarily develop singularity in finite time, unless the compression is strong enough for general data. In this paper, we identify a class of solutions of the full (non-isentropic) Euler equations, developing singularity in finite time even though their initial data do not contain any compression. This is in sharp contrast to the isentropic flow.
MSC:
| 76N15 | Gas dynamics (general theory) |
| 35L65 | Hyperbolic conservation laws |
| 35L67 | Shocks and singularities for hyperbolic equations |
References:
| [1] | Chemin, J., Remarques sur l’apparition de-singularités dans les ecoulements Euleriens compressibles, Comm. Math. Phys., 133, 323-329 (1990) · Zbl 0711.76076 |
| [2] | Chen, G., Formation of singularity and smooth wave propagation for the non-isentropic compressible Euler equations, J. Hyperbolic Differ. Equ., 8, 4, 671-690 (2011) · Zbl 1395.76036 |
| [5] | Chen, G.; Young, R.; Zhang, Q., Shock formation in the compressible Euler equations and related systems, J. Hyperbolic Differ. Equ., 10, 1, 149-172 (2013) · Zbl 1452.35137 |
| [6] | Christodoulou, D., The Formation of Shocks in 3-Dimensional Fluids, EMS Monographs in Mathematics (2007), European Mathematical Society (EMS): European Mathematical Society (EMS) Zürich · Zbl 1117.35001 |
| [7] | Christodoulou, D.; Miao, S., Compressible Flow and Euler’s Equations, Surveys of Modern Mathematics, vol. 9 (2014), International Press/Higher Education Press: International Press/Higher Education Press Somerville, MA/Beijing · Zbl 1329.76002 |
| [8] | Courant, R.; Friedrichs, K. O., Supersonic Flow and Shock Waves (1948), Wiley-Interscience: Wiley-Interscience New York · Zbl 0041.11302 |
| [9] | Dafermos, C., Hyperbolic Conservation Laws in Continuum Physics (2010), Springer-Verlag: Springer-Verlag Heidelberg · Zbl 1196.35001 |
| [10] | Grassin, M.; Serre, D., Global smooth solutions to Euler equations for an isentropic perfect gas, C. R. Acad. Sci. Paris Ser. I Math., 325, 721-726 (1997) · Zbl 0887.35125 |
| [11] | John, F., Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math., 27, 377-405 (1974) · Zbl 0302.35064 |
| [12] | Lax, P., Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys., 5, 5, 611-614 (1964) · Zbl 0135.15101 |
| [13] | Li, T.; Wang, D., Blowup phenomena of solutions to the Euler equations for compressible fluid flow, J. Differential Equations, 221, 91-101 (2006) · Zbl 1083.76051 |
| [14] | Li, T.; Zhou, Y.; Kong, D., Weak linear degeneracy and global classical solutions for general quasilinear hyperbolic systems, Comm. Partial Differential Equations, 19, 7-8, 1263-1317 (1994) · Zbl 0810.35054 |
| [15] | Lin, L.; Liu, H.; Yang, T., Existence of globally bounded continuous solutions for nonisentropic gas dynamics equations, J. Math. Anal. Appl., 209, 492-506 (1997) · Zbl 0880.35096 |
| [16] | Lin, L.; Vong, S., A note on the existence and nonexistence of globally bounded classical solutions for nonisentropic gas dynamics, Acta Math. Sci., 26B, 537-540 (2006) · Zbl 1147.35333 |
| [17] | Liu, F., Global smooth resolvability for one-dimensional gas dynamics systems, Nonlinear Anal., 36, 25-34 (1999) · Zbl 0923.35123 |
| [18] | Liu, T., The development of singularities in the nonlinear waves for quasi-linear hyperbolic partial differential equations, J. Differential Equations, 33, 92-111 (1979) · Zbl 0379.35048 |
| [19] | Makino, T.; Ukai, S.; Kawashima, S., Sur la solution \(\overset{``}{\text{a}}\) support compact de equations d”Euler compressible, Jpn. J. Appl. Math., 33, 249-257 (1986) · Zbl 0637.76065 |
| [20] | Rammaha, M. A., Formation of singularities in compressible fluids in two-space dimensions, Proc. Amer. Math. Soc., 107, 3, 705-714 (1989) · Zbl 0692.35015 |
| [21] | Serre, D., Classical global solutions of the Euler equations for a compressible perfect fluid, Ann. Inst. Fourier, 47, 139-153 (1997) · Zbl 0864.35069 |
| [22] | Sideris, T., Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys., 101, 475-485 (1985) · Zbl 0606.76088 |
| [23] | Smoller, J., Shock Waves and Reaction-Diffusion Equations (1982), Springer-Verlag: Springer-Verlag New York |
| [25] | Zheng, H., Singularity formation for the compressible Euler equations with general pressure law, J. Math. Anal. Appl., 438, 59-72 (2016) · Zbl 1334.35220 |
| [26] | Zhu, C., Global smooth solution of the nonisentropic gas dynamics system, Proc. Roy. Soc. Edinburgh Sect. A, 126, 768-775 (1996) · Zbl 0863.35084 |
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