Resonance for the isothermal system of isentropic gas dynamics

Author:
Yun-guang Lu

Journal:
Proc. Amer. Math. Soc. **139** (2011), 2821-2826

MSC (2010):
Primary 35L65, 76N10

Published electronically:
February 8, 2011

MathSciNet review:
2801623

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we remove the restriction or in the paper ``Existence of Solutions to Hyperbolic Conservation Laws with a Source'' (Commun. Math. Phys., 187 (1997), 327-340) and obtain the existence of solutions for the resonant, isothermal system of isentropic gas dynamics.

**[Di]**Ronald J. DiPerna,*Convergence of the viscosity method for isentropic gas dynamics*, Comm. Math. Phys.**91**(1983), no. 1, 1–30. MR**719807****[DW]**Xia Xi Ding and Jing Hua Wang,*Global solutions for a semilinear parabolic system*, Acta Math. Sci. (English Ed.)**3**(1983), no. 4, 397–414. MR**812546****[EGM]**Pedro Embid, Jonathan Goodman, and Andrew Majda,*Multiple steady states for 1-D transonic flow*, SIAM J. Sci. Statist. Comput.**5**(1984), no. 1, 21–41. MR**731879**, 10.1137/0905002**[Gl]**James Glimm,*Solutions in the large for nonlinear hyperbolic systems of equations*, Comm. Pure Appl. Math.**18**(1965), 697–715. MR**0194770****[GMP]**James Glimm, Guillermo Marshall, and Bradley Plohr,*A generalized Riemann problem for quasi-one-dimensional gas flows*, Adv. in Appl. Math.**5**(1984), no. 1, 1–30. MR**736548**, 10.1016/0196-8858(84)90002-2**[HW]**F.-M Huang and Z. Wang,*Convergence of viscosity solutions for isentropic gas dynamics*, SIAM J. Math. Anal.,**34**(2003), 595-610. (1984), 1-30.**[IT]**Eli Isaacson and Blake Temple,*Nonlinear resonance in systems of conservation laws*, SIAM J. Appl. Math.**52**(1992), no. 5, 1260–1278. MR**1182123**, 10.1137/0152073**[KL]**Christian Klingenberg and Yun-guang Lu,*Existence of solutions to hyperbolic conservation laws with a source*, Comm. Math. Phys.**187**(1997), no. 2, 327–340. MR**1463831**, 10.1007/s002200050138**[Liu]**Tai-Ping Liu,*Nonlinear resonance for quasilinear hyperbolic equation*, J. Math. Phys.**28**(1987), no. 11, 2593–2602. MR**913412**, 10.1063/1.527751**[Lu]**Y.-G. Lu,*Hyperbolic Conservation Laws and the Compensated Compactness Method*, Vol. 128, Chapman and Hall, CRC Press, New York, 2002.**[MMU]**Tetu Makino, Kiyoshi Mizohata, and Seiji Ukai,*The global weak solutions of compressible Euler equation with spherical symmetry*, Japan J. Indust. Appl. Math.**9**(1992), no. 3, 431–449. MR**1189949**, 10.1007/BF03167276**[Ni]**Takaaki Nishida,*Global solution for an initial boundary value problem of a quasilinear hyperbolic system*, Proc. Japan Acad.**44**(1968), 642–646. MR**0236526****[Ta]**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****[Ts1]**Naoki Tsuge,*Global 𝐿^{∞} solutions of the compressible Euler equations with spherical symmetry*, J. Math. Kyoto Univ.**46**(2006), no. 3, 457–524. MR**2311356****[Ts2]**Naoki Tsuge,*The compressible Euler equations for an isothermal gas with spherical symmetry*, J. Math. Kyoto Univ.**43**(2004), no. 4, 737–754. MR**2030796**

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Additional Information

**Yun-guang Lu**

Affiliation:
Department of Mathematics, Hangzhou Normal University, Hangzhou, 310036, People’s Republic of China – and – Department of Mathematics, National University of Colombia, Bogota, Colombia

Email:
yglu_2000@yahoo.com

DOI:
https://doi.org/10.1090/S0002-9939-2011-10733-0

Keywords:
Resonance,
gas dynamics,
global weak solution,
$L^{\infty}$ estimate

Received by editor(s):
April 6, 2010

Received by editor(s) in revised form:
April 27, 2010, and July 26, 2010

Published electronically:
February 8, 2011

Additional Notes:
The author thanks the referee for many valuable suggestions.

Communicated by:
Matthew J. Gursky

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.