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 Due to the subsonic property, the Lax entropy inequality [15] is no longer valid for subsonic phase transitions.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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

S. Benzoni-Gavage, Stability of subsonic planar phase boundaries in a van der Waals fluid, Arch. Rational Mech. Anal., 150 (1999), 23-55. 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), 43-62.  Google Scholar

[7]

R. Courant and K. O. Friedrichs, "Supersonic Flow and Shock Waves," Springer-Verlag, New York, 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), 941-1012. 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]

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.  Google Scholar

[11]

M. Grinfeld, Nonisothermal dynamic phase transitions, Quarterly Appl. Math., 47 (1989), 71-84.  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), 158-174. 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), 227-298.  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), 1-24. Google Scholar

[15]

P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math., 10 (1957), 537-467.  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), 153-197. 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, Birkhauser, 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), 525-549. doi: 10.1007/s002200050656.  Google Scholar

[19]

A. Majda, The stability of multi-dimensional shock fronts, Mem. Amer. Math. Soc., 275 (1983), 1-95.  Google Scholar

[20]

A. Majda, The existence of multi-dimensional shock fronts, Mem. Amer. Math. Soc., 281 (1983), 1-93.  Google Scholar

[21]

G. Métivier, Stability of multimensional shock fronts, in "Advances in The Theory of Shock Waves" 47 PnDEA, Birkhäuser, Boston, (2001), 25-103. Google Scholar

[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.  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), 45-49. 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), 301-315. doi: 10.1007/BF00250857.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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), 170-176. 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), 166-183. 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), 173-230. 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), 545-565. doi: 10.1017/S0308210500001773.  Google Scholar

[3]

S. Benzoni-Gavage, Nonuniqueness of phase transitions near the Maxwell line, Proc. Amer. Math. Soc., 127 (1999), 1183-1190. 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, T.M.A., 31 (1998), 243-263. 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), 23-55. 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), 43-62.  Google Scholar

[7]

R. Courant and K. O. Friedrichs, "Supersonic Flow and Shock Waves," Springer-Verlag, New York, 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), 941-1012. 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]

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.  Google Scholar

[11]

M. Grinfeld, Nonisothermal dynamic phase transitions, Quarterly Appl. Math., 47 (1989), 71-84.  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), 158-174. 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), 227-298.  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), 1-24. Google Scholar

[15]

P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math., 10 (1957), 537-467.  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), 153-197. 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, Birkhauser, 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), 525-549. doi: 10.1007/s002200050656.  Google Scholar

[19]

A. Majda, The stability of multi-dimensional shock fronts, Mem. Amer. Math. Soc., 275 (1983), 1-95.  Google Scholar

[20]

A. Majda, The existence of multi-dimensional shock fronts, Mem. Amer. Math. Soc., 281 (1983), 1-93.  Google Scholar

[21]

G. Métivier, Stability of multimensional shock fronts, in "Advances in The Theory of Shock Waves" 47 PnDEA, Birkhäuser, Boston, (2001), 25-103. Google Scholar

[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.  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), 45-49. 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), 301-315. doi: 10.1007/BF00250857.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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), 170-176. 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), 166-183. doi: 10.1137/S0036141097331056.  Google Scholar

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