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

Shu-Yi Zhang

doi:10.3934/dcds.2013.33.2221

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2013, Volume 33, Issue 5: 2221-2239. Doi: 10.3934/dcds.2013.33.2221

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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

Received: March 2011

Revised: September 2012

Published: May 2013

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Abstract

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 Due to the subsonic property, the Lax entropy inequality [15] is no longer valid for subsonic phase transitions.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.

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Mathematics Subject Classification: Primary: 35L45, 35L50; Secondary: 35L67.

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References

[1]	S. Alinhac, Existence d'ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels, Comm. Partial Differential Equaitons, 14 (1989), 173-230.doi: 10.1080/03605308908820595.
[2]	N. Bedjaoui and P. G. LeFloch, Diffusive-dispersive traveling waves and kinetic relations, Proc. Royal Soc. Edinburgh, 132 (2002), 545-565.doi: 10.1017/S0308210500001773.
[3]	S. Benzoni-Gavage, Nonuniqueness of phase transitions near the Maxwell line, Proc. Amer. Math. Soc., 127 (1999), 1183-1190.doi: 10.1090/S0002-9939-99-04719-X.
[4]	S. Benzoni-Gavage, Stability of multi-dimensional phase transitions in a van der Waals fluid, Nonlinear Analysis, T.M.A., 31 (1998), 243-263.doi: 10.1016/S0362-546X(96)00309-4.
[5]	S. Benzoni-Gavage, Stability of subsonic planar phase boundaries in a van der Waals fluid, Arch. Rational Mech. Anal., 150 (1999), 23-55.doi: 10.1007/s002050050179.
[6]	S. Chen, Global existence of supersonic flow past a curved convex wedge, J. Partial Differential Equation, 11 (1998), 43-62.
[7]	R. Courant and K. O. Friedrichs, "Supersonic Flow and Shock Waves," Springer-Verlag, New York, 1977.
[8]	J. F. Coulombel and P. Secchi, The stability of compressible vortex sheets in two space dimensions, Indiana Univ. Math. J., 53 (2004), 941-1012.doi: 10.1512/iumj.2004.53.2526.
[9]	J. F. Coulombel and P. Secchi, Nonlinear compressible vortex sheets in two space dimensions, Preprint.
[10]	H. Fan and M. Slemrod, Dynamic flows with liquid/vapor phase transitions, in "Handbook of Mathematical Fluid Dynamics" North-Holland, Amsterdam, (2002), 373-420.doi: 10.1016/S1874-5792(02)80011-8.
[11]	M. Grinfeld, Nonisothermal dynamic phase transitions, Quarterly Appl. Math., 47 (1989), 71-84.
[12]	H. Hattori, The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion nonisothermal case, J. Differential Equation, 65 (1986), 158-174.doi: 10.1016/0022-0396(86)90031-8.
[13]	H. O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., 23 (1970), 227-298.
[14]	D. J. Korteweg, Sur la forme que prennent leéquation des fluides si l'on tient compte des forces capilaires par des variantions densité, Arch. Néer. Sci. Exactes Sér., 2 (1901), 1-24.
[15]	P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math., 10 (1957), 537-467.
[16]	P. G. LeFloch, Propagating phase boundaries: formulation of the problem and existence via the Glimm method, Arch. Rational Mech. Anal., 123 (1993), 153-197.doi: 10.1007/BF00695275.
[17]	P. G. LeFloch, "Hyperbolic Systems of Conservation Laws. The Theory of Classical and Nonclassical Shock Waves," ETH Lectures in Mathematics, Birkhauser, 2002.doi: 10.1007/978-3-0348-8150-0.
[18]	W. Lien and T. P. Liu, Nonlinear stability of a self-similar 3-dimensional gas flow, Commun. Math. Phys., 204 (1999), 525-549.doi: 10.1007/s002200050656.
[19]	A. Majda, The stability of multi-dimensional shock fronts, Mem. Amer. Math. Soc., 275 (1983), 1-95.
[20]	A. Majda, The existence of multi-dimensional shock fronts, Mem. Amer. Math. Soc., 281 (1983), 1-93.
[21]	G. Métivier, Stability of multimensional shock fronts, in "Advances in The Theory of Shock Waves" 47 PnDEA, Birkhäuser, Boston, (2001), 25-103.
[22]	C. S. Morawetz, On a weak solution for transonic flow problem, Comm. Pure Appl. Math., 38 (1985), 423-443.doi: 10.1002/cpa.3160380610.
[23]	M. Shearer, Nonuniqueness of admissible solutions of Riemann initial value problem for a system of conservation laws of mixed type, Arch. Rational Mech. Anal., 93 (1986), 45-49.doi: 10.1007/BF00250844.
[24]	M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Arch. Rational Mech. Anal., 81 (1983), 301-315.doi: 10.1007/BF00250857.
[25]	M. Slemrod, Dynamic phase transitions in a van der waals fluid, J. Differential Equations, 52 (1984), 1-23.doi: 10.1016/0022-0396(84)90130-X.
[26]	Y.-G. Wang and Z. Xin, Stability and existence of multidimensional subsonic phase transitions, Acta Math. Appl. Sinica, 19 (2003), 529-558.doi: 10.1007/210255-003-0130-2.
[27]	S.-Y. Zhang, Existence of travelling waves in non-isothermal phase dynamics, J. Hyperbolic Differential Equations, 4 (2007), 391-400.doi: 10.1142/S0219891607001197.
[28]	S.-Y. Zhang, Stability of non-isothermal phase transitions in a steady van der waals flow, Preprint.
[29]	S.-Y. Zhang, Discontinuous solutions to the Euler equations in a van der Waals fluid, Appl. Math. Letters, 20 (2007), 170-176.doi: 10.1016/j.aml.2006.03.010.
[30]	Y. Zhang, Global existence of steady supersonic potential flow past a curved wedge with a piecewise smooth boundary, SIAM J. Math. Anal., 31 (1999), 166-183.doi: 10.1137/S0036141097331056.