American Institute of Mathematical Sciences

Advanced
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 Due to the subsonic property, the Lax entropy inequality [15] is no longer valid for subsonic phase transitions.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 - A, 2013, 33 (5) : 2221-2239. doi: 10.3934/dcds.2013.33.2221
References:
[1]

S. Alinhac, Existence d'ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels,, Comm. Partial Differential Equaitons, 14 (1989), 173.  doi: 10.1080/03605308908820595.  Google Scholar

[2]

N. Bedjaoui and P. G. LeFloch, Diffusive-dispersive traveling waves and kinetic relations,, Proc. Royal Soc. Edinburgh, 132 (2002), 545.  doi: 10.1017/S0308210500001773.  Google Scholar

[3]

S. Benzoni-Gavage, Nonuniqueness of phase transitions near the Maxwell line,, Proc. Amer. Math. Soc., 127 (1999), 1183.  doi: 10.1090/S0002-9939-99-04719-X.  Google Scholar

[4]

S. Benzoni-Gavage, Stability of multi-dimensional phase transitions in a van der Waals fluid,, Nonlinear Analysis, 31 (1998), 243.  doi: 10.1016/S0362-546X(96)00309-4.  Google Scholar

[5]

S. Benzoni-Gavage, Stability of subsonic planar phase boundaries in a van der Waals fluid,, Arch. Rational Mech. Anal., 150 (1999), 23.  doi: 10.1007/s002050050179.  Google Scholar

[6]

S. Chen, Global existence of supersonic flow past a curved convex wedge,, J. Partial Differential Equation, 11 (1998), 43.   Google Scholar

[7]

R. Courant and K. O. Friedrichs, "Supersonic Flow and Shock Waves,", Springer-Verlag, (1977).   Google Scholar

[8]

J. F. Coulombel and P. Secchi, The stability of compressible vortex sheets in two space dimensions,, Indiana Univ. Math. J., 53 (2004), 941.  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]

H. Fan and M. Slemrod, Dynamic flows with liquid/vapor phase transitions,, in, (2002), 373.  doi: 10.1016/S1874-5792(02)80011-8.  Google Scholar

[11]

M. Grinfeld, Nonisothermal dynamic phase transitions,, Quarterly Appl. Math., 47 (1989), 71.   Google Scholar

[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.  doi: 10.1016/0022-0396(86)90031-8.  Google Scholar

[13]

H. O. Kreiss, Initial boundary value problems for hyperbolic systems,, Comm. Pure Appl. Math., 23 (1970), 227.   Google Scholar

[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.   Google Scholar

[15]

P. D. Lax, Hyperbolic systems of conservation laws. II,, Comm. Pure Appl. Math., 10 (1957), 537.   Google Scholar

[16]

P. G. LeFloch, Propagating phase boundaries: formulation of the problem and existence via the Glimm method,, Arch. Rational Mech. Anal., 123 (1993), 153.  doi: 10.1007/BF00695275.  Google Scholar

[17]

P. G. LeFloch, "Hyperbolic Systems of Conservation Laws. The Theory of Classical and Nonclassical Shock Waves,", ETH Lectures in Mathematics, (2002).  doi: 10.1007/978-3-0348-8150-0.  Google Scholar

[18]

W. Lien and T. P. Liu, Nonlinear stability of a self-similar 3-dimensional gas flow,, Commun. Math. Phys., 204 (1999), 525.  doi: 10.1007/s002200050656.  Google Scholar

[19]

A. Majda, The stability of multi-dimensional shock fronts,, Mem. Amer. Math. Soc., 275 (1983), 1.   Google Scholar

[20]

A. Majda, The existence of multi-dimensional shock fronts,, Mem. Amer. Math. Soc., 281 (1983), 1.   Google Scholar

[21]

G. Métivier, Stability of multimensional shock fronts,, in, 47 (2001), 25.   Google Scholar

[22]

C. S. Morawetz, On a weak solution for transonic flow problem,, Comm. Pure Appl. Math., 38 (1985), 423.  doi: 10.1002/cpa.3160380610.  Google Scholar

[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.  doi: 10.1007/BF00250844.  Google Scholar

[24]

M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid,, Arch. Rational Mech. Anal., 81 (1983), 301.  doi: 10.1007/BF00250857.  Google Scholar

[25]

M. Slemrod, Dynamic phase transitions in a van der waals fluid,, J. Differential Equations, 52 (1984), 1.  doi: 10.1016/0022-0396(84)90130-X.  Google Scholar

[26]

Y.-G. Wang and Z. Xin, Stability and existence of multidimensional subsonic phase transitions,, Acta Math. Appl. Sinica, 19 (2003), 529.  doi: 10.1007/210255-003-0130-2.  Google Scholar

[27]

S.-Y. Zhang, Existence of travelling waves in non-isothermal phase dynamics,, J. Hyperbolic Differential Equations, 4 (2007), 391.  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]

S.-Y. Zhang, Discontinuous solutions to the Euler equations in a van der Waals fluid,, Appl. Math. Letters, 20 (2007), 170.  doi: 10.1016/j.aml.2006.03.010.  Google Scholar

[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.  doi: 10.1137/S0036141097331056.  Google Scholar

show all references

References:
[1]

S. Alinhac, Existence d'ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels,, Comm. Partial Differential Equaitons, 14 (1989), 173.  doi: 10.1080/03605308908820595.  Google Scholar

[2]

N. Bedjaoui and P. G. LeFloch, Diffusive-dispersive traveling waves and kinetic relations,, Proc. Royal Soc. Edinburgh, 132 (2002), 545.  doi: 10.1017/S0308210500001773.  Google Scholar

[3]

S. Benzoni-Gavage, Nonuniqueness of phase transitions near the Maxwell line,, Proc. Amer. Math. Soc., 127 (1999), 1183.  doi: 10.1090/S0002-9939-99-04719-X.  Google Scholar

[4]

S. Benzoni-Gavage, Stability of multi-dimensional phase transitions in a van der Waals fluid,, Nonlinear Analysis, 31 (1998), 243.  doi: 10.1016/S0362-546X(96)00309-4.  Google Scholar

[5]

S. Benzoni-Gavage, Stability of subsonic planar phase boundaries in a van der Waals fluid,, Arch. Rational Mech. Anal., 150 (1999), 23.  doi: 10.1007/s002050050179.  Google Scholar

[6]

S. Chen, Global existence of supersonic flow past a curved convex wedge,, J. Partial Differential Equation, 11 (1998), 43.   Google Scholar

[7]

R. Courant and K. O. Friedrichs, "Supersonic Flow and Shock Waves,", Springer-Verlag, (1977).   Google Scholar

[8]

J. F. Coulombel and P. Secchi, The stability of compressible vortex sheets in two space dimensions,, Indiana Univ. Math. J., 53 (2004), 941.  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]

H. Fan and M. Slemrod, Dynamic flows with liquid/vapor phase transitions,, in, (2002), 373.  doi: 10.1016/S1874-5792(02)80011-8.  Google Scholar

[11]

M. Grinfeld, Nonisothermal dynamic phase transitions,, Quarterly Appl. Math., 47 (1989), 71.   Google Scholar

[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.  doi: 10.1016/0022-0396(86)90031-8.  Google Scholar

[13]

H. O. Kreiss, Initial boundary value problems for hyperbolic systems,, Comm. Pure Appl. Math., 23 (1970), 227.   Google Scholar

[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.   Google Scholar

[15]

P. D. Lax, Hyperbolic systems of conservation laws. II,, Comm. Pure Appl. Math., 10 (1957), 537.   Google Scholar

[16]

P. G. LeFloch, Propagating phase boundaries: formulation of the problem and existence via the Glimm method,, Arch. Rational Mech. Anal., 123 (1993), 153.  doi: 10.1007/BF00695275.  Google Scholar

[17]

P. G. LeFloch, "Hyperbolic Systems of Conservation Laws. The Theory of Classical and Nonclassical Shock Waves,", ETH Lectures in Mathematics, (2002).  doi: 10.1007/978-3-0348-8150-0.  Google Scholar

[18]

W. Lien and T. P. Liu, Nonlinear stability of a self-similar 3-dimensional gas flow,, Commun. Math. Phys., 204 (1999), 525.  doi: 10.1007/s002200050656.  Google Scholar

[19]

A. Majda, The stability of multi-dimensional shock fronts,, Mem. Amer. Math. Soc., 275 (1983), 1.   Google Scholar

[20]

A. Majda, The existence of multi-dimensional shock fronts,, Mem. Amer. Math. Soc., 281 (1983), 1.   Google Scholar

[21]

G. Métivier, Stability of multimensional shock fronts,, in, 47 (2001), 25.   Google Scholar

[22]

C. S. Morawetz, On a weak solution for transonic flow problem,, Comm. Pure Appl. Math., 38 (1985), 423.  doi: 10.1002/cpa.3160380610.  Google Scholar

[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.  doi: 10.1007/BF00250844.  Google Scholar

[24]

M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid,, Arch. Rational Mech. Anal., 81 (1983), 301.  doi: 10.1007/BF00250857.  Google Scholar

[25]

M. Slemrod, Dynamic phase transitions in a van der waals fluid,, J. Differential Equations, 52 (1984), 1.  doi: 10.1016/0022-0396(84)90130-X.  Google Scholar

[26]

Y.-G. Wang and Z. Xin, Stability and existence of multidimensional subsonic phase transitions,, Acta Math. Appl. Sinica, 19 (2003), 529.  doi: 10.1007/210255-003-0130-2.  Google Scholar

[27]

S.-Y. Zhang, Existence of travelling waves in non-isothermal phase dynamics,, J. Hyperbolic Differential Equations, 4 (2007), 391.  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]

S.-Y. Zhang, Discontinuous solutions to the Euler equations in a van der Waals fluid,, Appl. Math. Letters, 20 (2007), 170.  doi: 10.1016/j.aml.2006.03.010.  Google Scholar

[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.  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]

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
[3]

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
[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]

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
[9]

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
[10]

Li Liu. Unique subsonic compressible potential flows in three -dimensional ducts. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 357-368. doi: 10.3934/dcds.2010.27.357
[11]

Gui-Qiang Chen, Jun Chen, Mikhail Feldman. Transonic flows with shocks past curved wedges for the full Euler equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4179-4211. doi: 10.3934/dcds.2016.36.4179
[12]

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
[13]

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
[14]

Pavel Drábek, Stephen Robinson. Continua of local minimizers in a quasilinear model of phase transitions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 163-172. doi: 10.3934/dcds.2013.33.163
[15]

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
[16]

Nicolai T. A. Haydn. Phase transitions in one-dimensional subshifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1965-1973. doi: 10.3934/dcds.2013.33.1965
[17]

Sylvie Benzoni-Gavage, Laurent Chupin, Didier Jamet, Julien Vovelle. On a phase field model for solid-liquid phase transitions. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 1997-2025. doi: 10.3934/dcds.2012.32.1997
[18]

Valeria Berti, Mauro Fabrizio, Diego Grandi. A phase field model for liquid-vapour phase transitions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 317-330. doi: 10.3934/dcdss.2013.6.317
[19]

João-Paulo Dias, Mário Figueira. On the Riemann problem for some discontinuous systems of conservation laws describing phase transitions. Communications on Pure & Applied Analysis, 2004, 3 (1) : 53-58. doi: 10.3934/cpaa.2004.3.53
[20]

Kelum Gajamannage, Erik M. Bollt. Detecting phase transitions in collective behavior using manifold's curvature. Mathematical Biosciences & Engineering, 2017, 14 (2) : 437-453. doi: 10.3934/mbe.2017027

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences