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

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 Due to the subsonic property, the Lax entropy inequality [15] is no longer valid for subsonic phase transitions.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

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

[1]

Martin Gugat, Michael Herty, Siegfried Müller. Coupling conditions for the transition from supersonic to subsonic fluid states. Networks & Heterogeneous Media, 2017, 12 (3) : 371-380. doi: 10.3934/nhm.2017016

[2]

Jan Prüss, Yoshihiro Shibata, Senjo Shimizu, Gieri Simonett. On well-posedness of incompressible two-phase flows with phase transitions: The case of equal densities. Evolution Equations & Control Theory, 2012, 1 (1) : 171-194. doi: 10.3934/eect.2012.1.171

[3]

Yanbo Hu, Tong Li. Sonic-supersonic solutions for the two-dimensional pseudo-steady full Euler equations. Kinetic & Related Models, 2019, 12 (6) : 1197-1228. doi: 10.3934/krm.2019046

[4]

José Luiz Boldrini, Luís H. de Miranda, Gabriela Planas. On singular Navier-Stokes equations and irreversible phase transitions. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2055-2078. doi: 10.3934/cpaa.2012.11.2055

[5]

Honghu Liu. Phase transitions of a phase field model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 883-894. doi: 10.3934/dcdsb.2011.16.883

[6]

Shaoqiang Tang, Huijiang Zhao. Stability of Suliciu model for phase transitions. Communications on Pure & Applied Analysis, 2004, 3 (4) : 545-556. doi: 10.3934/cpaa.2004.3.545

[7]

Tatyana S. Turova. Structural phase transitions in neural networks. Mathematical Biosciences & Engineering, 2014, 11 (1) : 139-148. doi: 10.3934/mbe.2014.11.139

[8]

Gui-Qiang Chen, Jun Chen, Mikhail Feldman. Transonic flows with shocks past curved wedges for the full Euler equations. Discrete & Continuous Dynamical Systems, 2016, 36 (8) : 4179-4211. doi: 10.3934/dcds.2016.36.4179

[9]

Holger Dullin, Yuri Latushkin, Robert Marangell, Shibi Vasudevan, Joachim Worthington. Instability of unidirectional flows for the 2D α-Euler equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2051-2079. doi: 10.3934/cpaa.2020091

[10]

Irena Lasiecka, Justin Webster. Eliminating flutter for clamped von Karman plates immersed in subsonic flows. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1935-1969. doi: 10.3934/cpaa.2014.13.1935

[11]

Eun Heui Kim. Boundary gradient estimates for subsonic solutions of compressible transonic potential flows. Conference Publications, 2007, 2007 (Special) : 573-579. doi: 10.3934/proc.2007.2007.573

[12]

Li Liu. Unique subsonic compressible potential flows in three -dimensional ducts. Discrete & Continuous Dynamical Systems, 2010, 27 (1) : 357-368. doi: 10.3934/dcds.2010.27.357

[13]

Guanming Gai, Yuanyuan Nie, Chunpeng Wang. A degenerate elliptic problem from subsonic-sonic flows in convergent nozzles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021070

[14]

Yanbo Hu, Jiequan Li. On a supersonic-sonic patch arising from the frankl problem in transonic flows. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021015

[15]

Steffen Arnrich. Modelling phase transitions via Young measures. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 29-48. doi: 10.3934/dcdss.2012.5.29

[16]

Paola Goatin. Traffic flow models with phase transitions on road networks. Networks & Heterogeneous Media, 2009, 4 (2) : 287-301. doi: 10.3934/nhm.2009.4.287

[17]

Pavel Drábek, Stephen Robinson. Continua of local minimizers in a quasilinear model of phase transitions. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 163-172. doi: 10.3934/dcds.2013.33.163

[18]

Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Isotropic-nematic phase transitions in liquid crystals. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 565-579. doi: 10.3934/dcdss.2011.4.565

[19]

Nicolai T. A. Haydn. Phase transitions in one-dimensional subshifts. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1965-1973. doi: 10.3934/dcds.2013.33.1965

[20]

Sylvie Benzoni-Gavage, Laurent Chupin, Didier Jamet, Julien Vovelle. On a phase field model for solid-liquid phase transitions. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 1997-2025. doi: 10.3934/dcds.2012.32.1997

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]