×

zbMATH — the first resource for mathematics

Formation of singularities in compressible fluids in two-space dimensions. (English) Zbl 0692.35015
The author studies the solution of the two-dimensional Euler equations for a polytropic ideal fluid; the associated Cauchy data are considered sufficiently regular to ensure the local existence of the solution and are constant outside a sphere of radius R. Moreover, the front of the initial disturbance satisfies certain positivity conditions. Under these assumptions, it is proved that any local \(C^ 1\)-solution of the considered problem, regardless of the size of the initial disturbance, will develop singularities in finite time.
Reviewer: I.Toma

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35Q05 Euler-Poisson-Darboux equations
35L45 Initial value problems for first-order hyperbolic systems
35L67 Shocks and singularities for hyperbolic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] R. Courant and D. Hilbert, Methods of mathematical physics, II, New York, Interscience, 1962. · Zbl 0099.29504
[2] Fritz John, Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math. 27 (1974), 377 – 405. · Zbl 0302.35064
[3] F. John, Blow-up of radial solutions of \?_{\?\?}=\?²(\?_{\?})\Delta \? in three space dimensions, Mat. Apl. Comput. 4 (1985), no. 1, 3 – 18 (English, with Portuguese summary).
[4] Tosio Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal. 58 (1975), no. 3, 181 – 205. · Zbl 0343.35056
[5] Sergiu Klainerman and Andrew Majda, Compressible and incompressible fluids, Comm. Pure Appl. Math. 35 (1982), no. 5, 629 – 651. · Zbl 0478.76091
[6] Tai Ping Liu, Development of singularities in the nonlinear waves for quasilinear hyperbolic partial differential equations, J. Differential Equations 33 (1979), no. 1, 92 – 111. · Zbl 0379.35048
[7] M. A. Rammaha, Finite-time blow-up for nonlinear wave equations in high dimensions, Comm. Partial Differential Equations 12 (1987), no. 6, 677 – 700. · Zbl 0631.35060
[8] M. A. Rammaha, On the blowing up of solutions to nonlinear wave equations in two space dimensions, J. Reine Angew. Math. 391 (1988), 55 – 64. · Zbl 0649.35062
[9] Thomas C. Sideris, Formation of singularities in solutions to nonlinear hyperbolic equations, Arch. Rational Mech. Anal. 86 (1984), no. 4, 369 – 381. · Zbl 0564.35070
[10] Thomas C. Sideris, Global behavior of solutions to nonlinear wave equations in three dimensions, Comm. Partial Differential Equations 8 (1983), no. 12, 1291 – 1323. · Zbl 0534.35069
[11] Thomas C. Sideris, Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys. 101 (1985), no. 4, 475 – 485. · Zbl 0606.76088
[12] Seiji Ukai, The incompressible limit and the initial layer of the compressible Euler equation, J. Math. Kyoto Univ. 26 (1986), no. 2, 323 – 331. · Zbl 0618.76074
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.