Abstract
We are concerned with globally defined entropy solutions to the Euler equations for compressible fluid flows in transonic nozzles with general cross-sectional areas. Such nozzles include the de Laval nozzles and other more general nozzles whose cross-sectional area functions are allowed at the nozzle ends to be either zero (closed ends) or infinity (unbounded ends). To achieve this, in this paper, we develop a vanishing viscosity method to construct globally defined approximate solutions and then establish essential uniform estimates in weighted Lp norms for the whole range of physical adiabatic exponents \({\gamma\in (1, \infty)}\), so that the viscosity approximate solutions satisfy the general Lp compensated compactness framework. The viscosity method is designed to incorporate artificial viscosity terms with the natural Dirichlet boundary conditions to ensure the uniform estimates. Then such estimates lead to both the convergence of the approximate solutions and the existence theory of globally defined finite-energy entropy solutions to the Euler equations for transonic flows that may have different end-states in the class of nozzles with general cross-sectional areas for all \({\gamma\in (1, \infty)}\). The approach and techniques developed here apply to other problems with similar difficulties. In particular, we successfully apply them to construct globally defined spherically symmetric entropy solutions to the Euler equations for all \({\gamma\in (1, \infty)}\).
Similar content being viewed by others
References
Chen, G.-Q.: Remarks on spherically symmetric solutions of the compressible Euler equations. Proc. R. Soc. Edinb. 127A, 243–259 (1997)
Chen, G.-Q.: Weak continuity and compactness for nonlinear partial differential equations. Chin. Ann. Math. 36B, 715–736 (2015)
Chen, G.-Q., Perepelitsa, M.: Vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow. Commun. Pure. Appl. Math. 63, 1469–1504 (2010)
Chen, G.-Q., Perepelitsa, M.: Vanishing viscosity solutions of the compressible Euler equations with spherical symmetry and large initial data. Commun. Math. Phys. 338, 771–800 (2015)
Courant, R., Friedrichs, K.O.: Supersonic Flow and Shock Waves. Springer, New York (1962)
Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, 4th edn. Springer, Berlin (2016)
DiPerna, R.J.: Convergence of the viscosity method for isentropic gas dynamics. Commun. Math. Phys. 91, 1–30 (1983)
Embid, P., Goodman, J., Majda, A.: Multiple steady states for 1-D transonic flow. SIAM J. Sci. Stat. Comput. 5, 21–41 (1984)
Germain, P., LeFloch, P.G.: Finite energy method for compressible fluids: the Navier-Stokes-Korteweg model. Commun. Pure Appl. Math. 69, 3–61 (2016)
Glaz, H., Liu, T.-P.: The asymptotic analysis of wave interactions and numerical calculations of transonic nozzle flow. Adv. Appl. Math. 5, 111–146 (1984)
Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math. 18, 697–715 (1965)
Glimm, J., Marshall, G., Plohr, B.: A generalized Riemann problem for quasi-one-dimensional gas flow. Adv. Appl. Math. 5, 1–30 (1984)
Guderley, G.: Starke kugelige und zylindrische Verdichtungsstösse in der Nähe des Kugelmittelpunktes bzw. der Zylinderachse. Luftfahrtforschung 19(9), 302–311 (1942)
Huang, F., Li, T., Yuan, D.: Global entropy solutions to multi-dimensional isentropic gas dynamics with spherical symmetry. Preprint arXiv:1711.04430 (2017)
LeFloch, P.G., Westdickenberg, M.: Finite energy solutions to the isentropic Euler equations with geometric effects. J. Math. Pures Appl. 88, 389–429 (2007)
Li, T., Wang, D.: Blowup phenomena of solutions to the Euler equations for compressible fluid flow. J. Differ. Equ. 221, 91–101 (2006)
Lions, P.-L., Perthame, B., Tadmor, E.: Kinetic formulation for the isentropic gas dynamics and p-system. Commun. Math. Phys. 163, 415–431 (1994)
Ladyzhenskaja, O.A., Solonnikov, V.A., Uraltseva, N.N.: Linear and Quasi-Linear Equations of Parabolic Type. LOMI-AMS, Providence (1968)
Liu, T.-P.: Quasilinear hyperbolic systems. Commun. Math. Phys. 68, 141–172 (1979)
Liu, T.-P.: Nonlinear stability and instability of transonic flows through a nozzle. Commun. Math. Phys. 83, 243–260 (1982)
Liu, T.-P.: Nonlinear resonance for quasilinear hyperbolic equation. J. Math. Phys. 28, 2593–2602 (1987)
Makino, T., Mizohata, K., Ukai, S.: The global weak solutions of compressible Euler equation with spherical symmetry. Jpn. J. Ind. Appl. Math. 9, 431–449 (1992)
Makino, T., Takeno, S.: Initial-boundary value problem for the spherically symmetric motion of isentropic gas. Jpn. J. Ind. Appl. Math. 11, 171–183 (1994)
Murat, F.: Compacité par compensation. Ann. Sc. Norm. Super. Pisa Sci. Fis. Mat. 5, 489–507 (1978)
Rosseland, S.: The Pulsation Theory of Variable Stars. Dover Publications Inc., New York (1964)
Slemrod, M.: Resolution of the spherical piston problem for compressible isentropic gas dynamics via a self-similar viscous limit. Proc. R. Soc. Edinb. 126A, 1309–1340 (1996)
Tartar, L.: Compensated compactness and applications to partial differential equations, In: Knops R.J. (ed.) Nonlinear Analysis and Mechanics, Herriot-Watt Symposium, Research Notes in Mathematics, vol. 4. Pitman Press (1979)
Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C. Dafermos
Rights and permissions
About this article
Cite this article
Chen, GQ.G., Schrecker, M.R.I. Vanishing Viscosity Approach to the Compressible Euler Equations for Transonic Nozzle and Spherically Symmetric Flows. Arch Rational Mech Anal 229, 1239–1279 (2018). https://doi.org/10.1007/s00205-018-1239-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-018-1239-z