Spherically symmetric solutions of the multidimensional, compressible, isentropic Euler equations
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- by Matthew R. I. Schrecker PDF
- Trans. Amer. Math. Soc. 373 (2020), 727-746 Request permission
Abstract:
In this paper, we prove the existence of finite-energy weak solutions to the compressible, isentropic Euler equations given arbitrary spherically symmetric initial data of finite energy. In particular, we show that the solutions to the spherically symmetric Euler equations obtained in recent works by Chen and Perepelitsa and Chen and Schrecker are weak solutions of the multidimensional, compressible Euler equations. This follows from new uniform estimates made on artificial viscosity approximations up to the origin, removing previous restrictions on admissible test functions and ruling out formation of an artificial boundary layer at the origin. The uniform estimates may be of independent interest concerning the possible rate of blowup of density and velocity at the origin for spherically symmetric flows.References
- Gui-Qiang Chen, Remarks on spherically symmetric solutions of the compressible Euler equations, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 2, 243–259. MR 1447952, DOI 10.1017/S0308210500023635
- Gui-Qiang Chen and Mikhail Perepelitsa, Vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow, Comm. Pure Appl. Math. 63 (2010), no. 11, 1469–1504. MR 2683391, DOI 10.1002/cpa.20332
- Gui-Qiang G. Chen and Mikhail Perepelitsa, Vanishing viscosity solutions of the compressible Euler equations with spherical symmetry and large initial data, Comm. Math. Phys. 338 (2015), no. 2, 771–800. MR 3351058, DOI 10.1007/s00220-015-2376-y
- Gui-Qiang G. Chen and Matthew R. I. Schrecker, Vanishing viscosity approach to the compressible Euler equations for transonic nozzle and spherically symmetric flows, Arch. Ration. Mech. Anal. 229 (2018), no. 3, 1239–1279. MR 3814602, DOI 10.1007/s00205-018-1239-z
- R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, N. Y., 1948. MR 0029615
- G. Guderley, Starke kugelige und zylindrische Verdichtungsstösse in der Nähe des Kugelmittelpunktes bzw. der Zylinderachse, Luftfahrtforschung 19 (1942), 302–311 (German). MR 8522
- David Hoff, Spherically symmetric solutions of the Navier-Stokes equations for compressible, isothermal flow with large, discontinuous initial data, Indiana Univ. Math. J. 41 (1992), no. 4, 1225–1302. MR 1206346, DOI 10.1512/iumj.1992.41.41060
- Helge Kristian Jenssen and Charis Tsikkou, On similarity flows for the compressible Euler system, J. Math. Phys. 59 (2018), no. 12, 121507, 25. MR 3894017, DOI 10.1063/1.5049093
- Philippe G. LeFloch and Michael Westdickenberg, Finite energy solutions to the isentropic Euler equations with geometric effects, J. Math. Pures Appl. (9) 88 (2007), no. 5, 389–429 (English, with English and French summaries). MR 2369876, DOI 10.1016/j.matpur.2007.07.004
- P.-L. Lions, B. Perthame, and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and $p$-systems, Comm. Math. Phys. 163 (1994), no. 2, 415–431. MR 1284790, DOI 10.1007/BF02102014
- François Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 3, 489–507 (French). MR 506997
- L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass.-London, 1979, pp. 136–212. MR 584398
Additional Information
- Matthew R. I. Schrecker
- Affiliation: University of Wisconsin–Madison, Van Vleck Hall, 480 Lincoln Drive, Madison, Wisconsin 53706
- MR Author ID: 1275086
- Email: schrecker@wisc.edu
- Received by editor(s): February 7, 2019
- Received by editor(s) in revised form: August 1, 2019, August 25, 2019, and August 26, 2019
- Published electronically: October 1, 2019
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 727-746
- MSC (2010): Primary 35Q35, 35Q31, 35B44, 35L65, 76N10
- DOI: https://doi.org/10.1090/tran/7980
- MathSciNet review: 4042890