Blow-up phenomena of solutions to the Euler equations for compressible fluid flow. (English) Zbl 1083.76051

Summary: The blow-up phenomena are investigated for Euler equations of compressible fluid flow. The approach is to construct special explicit solutions with spherical symmetry to study certain blow-up behavior of multi-dimensional solutions. In particular, the special solutions with velocity of the form \(c(t)\mathbf x\) are constructed to show the expanding and blow-up properties. The solution with velocity of the form \(\dot a(t)\mathbf x/a(t)\) for \(\gamma \geqslant 1\) and for any space dimensions is obtained as a corollary. Another conclusion is that there is only trivial solution with velocity of the form \(c(t)|\mathbf x|^{\alpha -1}\mathbf x\) for \(\alpha\neq 1\) and multi-space dimensions.


76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q35 PDEs in connection with fluid mechanics
Full Text: DOI


[1] Alinhac, S., Blowup for nonlinear hyperbolic equations, (1995), Birkhauser Boston · Zbl 0820.35001
[2] Barenblatt, G.I., ()
[3] Chemin, J.Y., Remarques sur l’apparition de-singularités dans LES ecoulements euleriens compressibles, Comm. math. phys., 133, 323-329, (1990) · Zbl 0711.76076
[4] Chen, G.-Q., Remarks on spherically symmetric solutions of the compressible Euler equations, Proc. roy. soc. Edinburgh, 127A, 243-259, (1997) · Zbl 0877.35100
[5] Chen, G.-Q.; Chen, S.-X.; Wang, D.; Wang, Z., A multidimensional piston problem for the Euler equations for compressible flow, Discrete and continuous dynamical systems, 13, 361-383, (2005) · Zbl 1077.35090
[6] Chen, G.-Q.; Glimm, J., Global solution to the compressible Euler equations with geometrical structure, Comm. math. phys., 179, 153-193, (1996) · Zbl 0857.76073
[7] Chen, G.-Q.; Li, T.H., Global entropy solutions in \(L^\infty\) to the Euler equations and euler – poisson equations for isothermal fluids with spherical symmetry, Methods appl. anal., 10, 2, 215-243, (2003) · Zbl 1059.35094
[8] Chen, G.-Q.; Wang, D., (), 421-543
[9] Courant, R.; Friedrichs, K.O., Supersonic flow and shock waves, (1962), Springer New York · Zbl 0365.76001
[10] Dafermos, C.M., Hyperbolic conservation laws in continuum physics, (2000), Springer Berlin · Zbl 0940.35002
[11] Deng, Y.; Liu, T.-P.; Yang, T.; Yao, Z.-A., Solutions of euler – poisson equations for gaseous stars, Arch. rational mech. anal., 164, 261-285, (2002) · Zbl 1038.76036
[12] Deng, Y.; Xiang, J.; Yang, T., Blowup phenomena of solutions to euler – poisson equations, J. math. anal. appl., 286, 295-306, (2003) · Zbl 1032.35023
[13] Fu, C.-C.; Lin, S.-S., On the critical mass of the collapse of a gaseous star in spherically symmetric and isentropic motion, Japan J. ind. appl. math., 15, 461-469, (1998) · Zbl 0913.35108
[14] Landau, L.D.; Lifshitz, E.M., Fluid mechanics, (1987), Butterworth-Heinemann Oxford · Zbl 0146.22405
[15] Li, T.H., Some special solutions of the Euler equation in \(\mathbb{R}^N\), Comm. pure appl. anal., 4, 757-762, (2005) · Zbl 1083.35058
[16] Liu, T.P., Compressible flow with damping and vacuum, Japan J. indust. appl. math., 13, 25-32, (1996) · Zbl 0865.35107
[17] Majda, A., ()
[18] Makino, T., Blowing up solutions of the euler – poisson equation for the evolution of gaseous stars, Proceedings of the fourth international workshop on mathematical aspects of fluid and plasma dynamics (Kyoto, 1991), Transport theory statist. phys., 21, 615-624, (1992) · Zbl 0793.76069
[19] Sedov, L.I., Similarity and dimensional methods in mechanics, transl. by M. Friedman (transl. M. Holt (ed.),), (1959), Academic Press New York, London · Zbl 0121.18504
[20] Sideris, T.C., Formation of singularities in three-dimensional compressible fluids, Comm. math. phys., 101, 475-485, (1985) · Zbl 0606.76088
[21] G.I. Taylor, The Formation of a Blast Wave by a Very Intense Explosion, Ministry of Home Security RC 210 (II-5-153) 1941.
[22] G.I. Taylor, The Propagation and Decay of Blast Waves, British Civilian Defense Research Committee, 1944.
[23] Whitham, G.B., Linear and nonlinear waves, (1973), Wiley-Interscience New York · Zbl 0373.76001
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