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May  2013, 33(5): 2221-2239. doi: 10.3934/dcds.2013.33.2221

Existence of multidimensional non-isothermal phase transitions in a steady van der Waals flow

Shu-Yi Zhang 1,

1. 

Business information management school, Shanghai institute of foreign trade, 1900 Wenxiang Rd., Shanghai 201620, China

Received  March 2011 Revised  September 2012 Published  December 2012

This paper is concerned with the existence of multi-dimensional non-isothermal subsonic phase transitions in a steady supersonic flow with the van der Waals type state function. Due to the subsonic property, the Lax entropy inequality [15] is no longer valid for subsonic phase transitions. Hence, physical admissible planar waves are chosen by the viscosity capillarity criterion [24]. Based on the uniform stability result in [28], we perform the iteration scheme [20] and establish the existence.
Keywords: Supersonic flows, Euler equations., subsonic phase transitions.
Mathematics Subject Classification: Primary: 35L45, 35L50; Secondary: 35L6.
Citation: Shu-Yi Zhang. Existence of multidimensional non-isothermal phase transitions in a steady van der Waals flow. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 2221-2239. doi: 10.3934/dcds.2013.33.2221
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show all references

References:
[1]

Comm. Partial Differential Equaitons, 14 (1989), 173-230. doi: 10.1080/03605308908820595.  Google Scholar

[2]

Proc. Royal Soc. Edinburgh, 132 (2002), 545-565. doi: 10.1017/S0308210500001773.  Google Scholar

[3]

Proc. Amer. Math. Soc., 127 (1999), 1183-1190. doi: 10.1090/S0002-9939-99-04719-X.  Google Scholar

[4]

Nonlinear Analysis, T.M.A., 31 (1998), 243-263. doi: 10.1016/S0362-546X(96)00309-4.  Google Scholar

[5]

Arch. Rational Mech. Anal., 150 (1999), 23-55. doi: 10.1007/s002050050179.  Google Scholar

[6]

J. Partial Differential Equation, 11 (1998), 43-62.  Google Scholar

[7]

Springer-Verlag, New York, 1977.  Google Scholar

[8]

Indiana Univ. Math. J., 53 (2004), 941-1012. doi: 10.1512/iumj.2004.53.2526.  Google Scholar

[9]

J. F. Coulombel and P. Secchi, Nonlinear compressible vortex sheets in two space dimensions,, Preprint., ().   Google Scholar

[10]

in "Handbook of Mathematical Fluid Dynamics" North-Holland, Amsterdam, (2002), 373-420. doi: 10.1016/S1874-5792(02)80011-8.  Google Scholar

[11]

Quarterly Appl. Math., 47 (1989), 71-84.  Google Scholar

[12]

J. Differential Equation, 65 (1986), 158-174. doi: 10.1016/0022-0396(86)90031-8.  Google Scholar

[13]

Comm. Pure Appl. Math., 23 (1970), 227-298.  Google Scholar

[14]

Arch. Néer. Sci. Exactes Sér., 2 (1901), 1-24. Google Scholar

[15]

Comm. Pure Appl. Math., 10 (1957), 537-467.  Google Scholar

[16]

Arch. Rational Mech. Anal., 123 (1993), 153-197. doi: 10.1007/BF00695275.  Google Scholar

[17]

ETH Lectures in Mathematics, Birkhauser, 2002. doi: 10.1007/978-3-0348-8150-0.  Google Scholar

[18]

Commun. Math. Phys., 204 (1999), 525-549. doi: 10.1007/s002200050656.  Google Scholar

[19]

Mem. Amer. Math. Soc., 275 (1983), 1-95.  Google Scholar

[20]

Mem. Amer. Math. Soc., 281 (1983), 1-93.  Google Scholar

[21]

in "Advances in The Theory of Shock Waves" 47 PnDEA, Birkhäuser, Boston, (2001), 25-103. Google Scholar

[22]

Comm. Pure Appl. Math., 38 (1985), 423-443. doi: 10.1002/cpa.3160380610.  Google Scholar

[23]

Arch. Rational Mech. Anal., 93 (1986), 45-49. doi: 10.1007/BF00250844.  Google Scholar

[24]

Arch. Rational Mech. Anal., 81 (1983), 301-315. doi: 10.1007/BF00250857.  Google Scholar

[25]

J. Differential Equations, 52 (1984), 1-23. doi: 10.1016/0022-0396(84)90130-X.  Google Scholar

[26]

Acta Math. Appl. Sinica, 19 (2003), 529-558. doi: 10.1007/210255-003-0130-2.  Google Scholar

[27]

J. Hyperbolic Differential Equations, 4 (2007), 391-400. doi: 10.1142/S0219891607001197.  Google Scholar

[28]

S.-Y. Zhang, Stability of non-isothermal phase transitions in a steady van der waals flow,, Preprint., ().   Google Scholar

[29]

Appl. Math. Letters, 20 (2007), 170-176. doi: 10.1016/j.aml.2006.03.010.  Google Scholar

[30]

SIAM J. Math. Anal., 31 (1999), 166-183. doi: 10.1137/S0036141097331056.  Google Scholar

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