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 Due to the subsonic property, the Lax entropy inequality [15] is no longer valid for subsonic phase transitions.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 and Continuous Dynamical Systems, 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), 173-230. 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.

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), 173-230. 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.

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