PDF
Abstract
In this paper, the author studies the multidimensional stability of subsonic phase transitions in a steady supersonic flow of van der Waals type. The viscosity capillarity criterion (in “Arch. Rat. Mech. Anal., 81(4), 1983, 301–315”) is used to seek physical admissible planar waves. By showing the Lopatinski determinant being non-zero, it is proved that subsonic phase transitions are uniformly stable in the sense of Majda (in “Mem. Amer. Math. Soc., 41(275), 1983, 1–95”) under both one dimensional and multidimensional perturbations.
Keywords
Supersonic flows
/
Subsonic phase transitions
/
Euler equations
/
Multi-dimensional stability
Cite this article
Download citation ▾
Shuyi Zhang.
Stability of multidimensional phase transitions in a steady van der Waals flow.
Chinese Annals of Mathematics, Series B, 2008, 29(3): 223-238 DOI:10.1007/s11401-007-0242-8
| [1] |
Benzoni-Gavage S.. Stability of multi-dimensional phase transitions in a van der Waals fluid. Nonlinear Anal., 1998, 31(1–2): 243-263
|
| [2] |
Benzoni-Gavage S.. Stability of subsonic planar phase boundaries in a van der Waals fluid. Arch. Rat. Mech. Anal., 1999, 150(1): 23-55
|
| [3] |
Chen S.. Global existence of supersonic flow past a curved convex wedge. J. Partial Diff. Eqs, 1998, 11: 43-62
|
| [4] |
Courant R., Friedrichs K. O.. Supersonic Flow and Shock Waves, 1977, New York: Springer-Verlag
|
| [5] |
Fan H., Slemrod M.. Dynamic flows with liquid/vapor phase transitions, Handbook of Mathematical Fluid Dynamics, Vol. I, 2002, Amsterdam: North-Holland 373-420
|
| [6] |
Kreiss H. O.. Initial boundary value problems for hyperbolic systems. Comm. Pure Appl. Math., 1970, 23: 227-298
|
| [7] |
Lax P. D.. Hyperbolic systems of conservation laws. II. Comm. Pure Appl. Math., 1957, 10: 537-467
|
| [8] |
LeFloch P. G.. Propagating phase boundaries: formulation of the problem and existence via the Glimm method. Arch. Rat. Mech. Anal., 1993, 123(2): 153-197
|
| [9] |
Li T., Wang L.. Inverse piston problem for the system of one-dimensional isentropic flow. Chin. Ann. Math., 2007, 28B(3): 265-282
|
| [10] |
Lien W., Liu T. P.. Nonlinear stability of a self-similar 3-dimensional gas flow. Commun. Math. Phys., 1999, 204: 525-549
|
| [11] |
Majda A.. The stability of multi-dimensional shock fronts. Mem. Amer. Math. Soc., 1983, 41(275): 1-95
|
| [12] |
Morawetz C. S.. On a weak solution for transonic flow problem. Comm. Pure Appl. Math., 1985, 38: 423-443
|
| [13] |
Shearer M.. Nonuniqueness of admissible solutions of Riemann initial value problem for a system of conservation laws of mixed type. Arch. Rat. Mech. Anal., 1986, 93(1): 45-59
|
| [14] |
Slemrod M.. Admissibility criteria for propagating phase boundaries in a van der Waals fluid. Arch. Rat. Mech. Anal., 1983, 81(4): 301-315
|
| [15] |
Wang Y. G., Xin Z.. Stability and existence of multidimensional subsonic phase transitions. Acta Math. Appl. Sin. Engl. Ser., 2003, 19(4): 529-558
|
| [16] |
Zhang S. Y.. Discontinuous solutions to the Euler equations in a van der Waals fluid. Appl. Math. Letters, 2007, 20(2): 170-176
|
| [17] |
Zhang S. Y.. Existence of travelling waves in non-isothermal phase dynamics. J. Hyperbolic Diff. Eqs., 2007, 4(3): 391-400
|
| [18] |
Zhang, S. Y., Stability of Phase Transitions in a Steady van der Waals Fluid, preprint.
|
| [19] |
Zhang Y.. Global existence of steady supersonic potential flow past a curved wedge with a piecewise smooth boundary. SIAM J. Math. Anal., 1999, 31: 166-183
|
