×

zbMATH — the first resource for mathematics

Global solutions to the compressible Euler equations with geometrical structure. (English) Zbl 0857.76073
Summary: We prove the existence of global solutions to the Euler equations of compressible isentropic gas dynamics with geometrical structure, including transonic nozzle flow and spherically symmetric flow. Due to the presence of the geometrical source terms, the existence results themselves are new, especially as they pertain to radial flow in an unbounded region, \(|\vec x|\geq 1\), and to transonic nozzle flow. Arbitrary data with \(L^\infty\) bounds are allowed in these results. A shock capturing numerical scheme is introduced to compute such flows and to construct approximate solutions. The convergence and consistency of the approximate solutions generated from this scheme to the global solutions are proved with the aid of a compensated compactness framework.

MSC:
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q35 PDEs in connection with fluid mechanics
76M20 Finite difference methods applied to problems in fluid mechanics
76H05 Transonic flows
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] [C1] Chen, G.-Q.: The compensated compactness method and the system of isentropic gas dynamics. MSRI Preprint 00527-91, Berkeley (1990)
[2] [C2] Chen, G.-Q.: Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics (III). Acta Mathematica Scientia8, 243–276 (1988) (in Chinese);6, 75–120 (1986) (in English)
[3] [C3] Chen, G.-Q.: Remarks on spherically symmetric solutions to the compressible Euler equations. Proc. Royal Soc. Edinburg, 1996 (to appear)
[4] [CG] Chen, G.-Q. and Glimm, J.: Global solutions to the cylindrically symmetric rotating motion of isentropic gas. Z. Angew. Math. Phys. 1996 (to appear)
[5] [CW] Chen, G.-Q. and Wang, D.: Convergence of shock capturing schemes for the compressible Euler-Poisson equations. Commun. Math. Phys. 1996 (to appear) · Zbl 0858.76051
[6] [CF] Courant, R. and Friedrichs, K.O.: Supersonic flow and shock waves. New York, Springer-Verlag, 1948 · Zbl 0041.11302
[7] [D1] Dafermos, C.: Polygonal approximations of solutions of the initial-value problem for a conservation law. J. Math. Anal. Appl.38, 33–41 (1972) · Zbl 0233.35014 · doi:10.1016/0022-247X(72)90114-X
[8] [D2] Dafermos, C.: Hyperbolic Systems of Conservation Laws. In: Systems of Nonlinear Partial Differential Equations, NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci.111, 25–70 (1983)
[9] [DC1] Ding, X., Chen, G.-Q. and Luo, P.: Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics (I)–(II). Acta Mathematica Scientia7, 467–480 (1987),8 61–94 (1988) (in Chinese);5, 415–432, 433–472 (in English)
[10] [DC2] Ding, X., Chen, G.-Q. and Luo, P.: Convergence of the fractional step Lax-Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics. Commun. Math. Phys.121, 63–84 (1989) · Zbl 0689.76022 · doi:10.1007/BF01218624
[11] [Di] DiPerna, R.: Convergence of the viscosity method for isentropic gas dynamics. Commun. Math. Phys.91, 1–30 (1983) · Zbl 0533.76071 · doi:10.1007/BF01206047
[12] [EM] Embid, P., Goodman, J. and Majda, A.: Multiple steady state for 1-D transonic flow. SIAM J. Sci. Stat. Comp.5, 21–41 (1984) · Zbl 0573.76055 · doi:10.1137/0905002
[13] [Fo] Fok, S.K.: Extensions of Glimm’s method to the problem of gas flow in a duct of variable cross-section. Ph.D. Thesis, Department of Mathematics, University of California at Berkeley, 1980
[14] [GL] Glaz, H. and Liu, T.-P.: The asymptotic analysis of wave interactions and numerical calculation of transonic nozzle flow. Adv. Appl. Math.5, 111–146 (1984) · Zbl 0598.76065 · doi:10.1016/0196-8858(84)90006-X
[15] [Gl] Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Commun. Math. Phys.18, 697–715 (1965) · Zbl 0141.28902
[16] [GM] Glimm, J., Marshall, G. and Plohr, B.: A generalized Riemann problem for quasi-one-dimensional gas flow. Adv. Appl. Math.5, 1–30 (1984) · Zbl 0566.76056 · doi:10.1016/0196-8858(84)90002-2
[17] [Ho] Hoff, D.: Spherically symmetric solutions of the Navier-Stokes equations for compressible, isothermal flow with large, discontinuous initial data. Indiana U. Math. J.41, 1225–1302 (1992) · Zbl 0765.35033 · doi:10.1512/iumj.1992.41.41060
[18] [La1] Lax, P.D.: Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. CBMS.11, SIAM, 1973
[19] [La2] Lax, P.D.: Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun. Pure Appl. Math.7, 159–193 (1954) · Zbl 0055.19404 · doi:10.1002/cpa.3160070112
[20] [LP] Lions, P.L., Perthame, B. and Tadmor, E.: Kinetic formulation of the isentropic gas dynamics andp-systems. Commun. Math. Phys.163, 169–192 (1994) · Zbl 0799.35151 · doi:10.1007/BF02102014
[21] [L1] Liu, T.-P.: Quasilinear hyperbolic systems. Commun. Math. Phys.68, 141–172 (1979) · Zbl 0435.35054 · doi:10.1007/BF01418125
[22] [L2] Liu, T.-P.: Nonlinear stability and instability of trasonic gas flow through a nozzle. Commun. Math. Phys.83, 243–260 (1983) · Zbl 0576.76053 · doi:10.1007/BF01976043
[23] [L3] Liu, T.-P.: Nonlinear resonance for quasilinear hyperbolic equation. J. Math. Phys.28, 2593–2602 (1987) · Zbl 0662.35068 · doi:10.1063/1.527751
[24] [MU] Makino, T., Mizohata, K. and Ukai, S.: Global weak solutions of the compressible Euler equations with spherical symmetry I, II. Japan J. Industrial Appl. Math.9, 431–449 (1992) · Zbl 0761.76085 · doi:10.1007/BF03167276
[25] [MT] Makino, T. and Takeno, S.: Initial-Boundary Value Problem for the spherically symmetric motion of isentropic gas. Japan J. Industrial Appl. Math.11, 171–183 (1994) · Zbl 0797.76077 · doi:10.1007/BF03167220
[26] [M] Morawetz, C.S.: On a weak solution for a transonic flow problem. Commun. Pure Appl. Math.38, 797–818 (1985) · Zbl 0615.76070 · doi:10.1002/cpa.3160380610
[27] [Mu] Murat, F.: L’injection du cone positif deH dansW ,q est compacte pour toutq<2. J. Math. Pures Appl.60, 309–322 (1981)
[28] [OM] Okada, M. and Makino, T.: Free boundary problem for the equation of spherically symmetric motion of viscous gas. Japan J. Indust. Appl. Math. (to appear) (1994) · Zbl 0783.76082
[29] [Se] Serre, D.: La compacité par compensation pour les systèmes hyperboliques non linéaires de deux équations à une dimension d’espace. J. Math. Pures Appl.65, 423–468 (1986) · Zbl 0601.35070
[30] [Sl] Slemrod, M.: Resolution of the spherical piston problem for compressible isentropic gas dynamics via a self-similar viscous limit. Proc. Royal Soc. Edinburgh 1996 (to appear) · Zbl 0866.76075
[31] [Sm] Smoller, J.: Shock Waves and Reaction-Diffusion Equations. New York: Springer-Verlag, 1983 · Zbl 0508.35002
[32] [Ta] Tartar, L.: Compensated compactness and applications to partial differential equations. Research Notes in Mathematics, Nonlinear Analysis and Mechanics, ed. R.J. Knops, Vol.4, New York: Pitman Press, 1979 · Zbl 0437.35004
[33] [Wh] Whitham, G.B.: Linear and Nonlinear Waves. New York: John Wiley and Sons, 1974 · Zbl 0373.76001
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.