The uniqueness and stability of the solution of the Riemann problem of a system of conservation laws of mixed type

Author:
Hai Tao Fan

Journal:
Trans. Amer. Math. Soc. **333** (1992), 913-938

MSC:
Primary 35L65; Secondary 76L05

MathSciNet review:
1104200

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Abstract | References | Similar Articles | Additional Information

Abstract: We establish the uniqueness and stability of the similarity solution of the Riemann problem for a system of conservation laws of mixed type, with initial data separated by the elliptic region, which satisfies the viscosity-capillarity travelling wave admissibility criterion.

**[1]**Constantine M. Dafermos,*The entropy rate admissibility criterion for solutions of hyperbolic conservation laws*, J. Differential Equations**14**(1973), 202–212. MR**0328368****[2]**Constantine M. Dafermos,*Solution of the Riemann problem for a class of hyperbolic systems of conservation laws by the viscosity method*, Arch. Rational Mech. Anal.**52**(1973), 1–9. MR**0340837****[3]**-,*Structure of the solutions of the Riemann problem for hyperbolic conservation laws*, Arch. Rational Mech. Anal.**53**(1974), 203-217.**[4]**C. M. Dafermos and R. J. DiPerna,*The Riemann problem for certain classes of hyperbolic systems of conservation laws*, J. Differential Equations**20**(1976), no. 1, 90–114. MR**0404871****[5]**C. M. Dafermos,*Admissible wave fans in nonlinear hyperbolic systems*, Arch. Rational Mech. Anal.**106**(1989), no. 3, 243–260. MR**981663**, 10.1007/BF00281215**[6]**Hai Tao Fan,*The structure of solutions of gas dynamic equations and the formation of the vacuum state*, Quart. Appl. Math.**49**(1991), no. 1, 29–48. MR**1096230****[7]**Hai Tao Fan,*A limiting “viscosity” approach to the Riemann problem for materials exhibiting a change of phase. II*, Arch. Rational Mech. Anal.**116**(1992), no. 4, 317–337. MR**1132765**, 10.1007/BF00375671**[8]**R. Hagan and M. Slemrod,*The viscosity-capillarity criterion for shocks and phase transitions*, Arch. Rational Mech. Anal.**83**(1983), no. 4, 333–361. MR**714979**, 10.1007/BF00963839**[9]**L. Hsiao,*Admissibility criteria and admissible weak solutions of Riemann problems for conservation laws of mixed type: a summary*, Nonlinear evolution equations that change type, IMA Vol. Math. Appl., vol. 27, Springer, New York, 1990, pp. 85–88. MR**1074187**, 10.1007/978-1-4613-9049-7_7**[10]**-,*Uniqueness of admissible solutions of Riemann problem of system of conservation laws of mixed type*, J. Differential Equations (to appear).**[11]**Richard D. James,*The propagation of phase boundaries in elastic bars*, Arch. Rational Mech. Anal.**73**(1980), no. 2, 125–158. MR**556559**, 10.1007/BF00258234**[12]**A. S. Kalašnikov,*Construction of generalized solutions of quasi-linear equations of first order without convexity conditions as limits of solutions of parabolic equations with a small parameter*, Dokl. Akad. Nauk SSSR**127**(1959), 27–30 (Russian). MR**0108651****[13]**P. D. Lax,*Hyperbolic systems of conservation laws. II*, Comm. Pure Appl. Math.**10**(1957), 537–566. MR**0093653****[14]**Tai Ping Liu,*The Riemann problem for general systems of conservation laws*, J. Differential Equations**18**(1975), 218–234. MR**0369939****[15]**Michael Shearer,*The Riemann problem for a class of conservation laws of mixed type*, J. Differential Equations**46**(1982), no. 3, 426–443. MR**681232**, 10.1016/0022-0396(82)90103-6**[16]**Michael Shearer,*Nonuniqueness of admissible solutions of Riemann initial value problems for a system of conservation laws of mixed type*, Arch. Rational Mech. Anal.**93**(1986), no. 1, 45–59. MR**822335**, 10.1007/BF00250844**[17]**Michael Shearer,*Dynamic phase transitions in a van der Waals gas*, Quart. Appl. Math.**46**(1988), no. 4, 631–636. MR**973380****[18]**M. Slemrod,*Admissibility criteria for propagating phase boundaries in a van der Waals fluid*, Arch. Rational Mech. Anal.**81**(1983), no. 4, 301–315. MR**683192**, 10.1007/BF00250857**[19]**M. Slemrod,*Dynamics of first order phase transitions*, Phase transformations and material instabilities in solids (Madison, Wis., 1983) Publ. Math. Res. Center Univ. Wisconsin, vol. 52, Academic Press, Orlando, FL, 1984, pp. 163–203. MR**802225****[20]**M. Slemrod,*A limiting “viscosity” approach to the Riemann problem for materials exhibiting change of phase*, Arch. Rational Mech. Anal.**105**(1989), no. 4, 327–365. MR**973246**, 10.1007/BF00281495**[21]**M. Slemrod and A. Tzavaras,*A limiting viscosity approach for the Riemann problem in isentropic gas dynamics*, Indiana Univ. Math. J. (submitted).**[22]**M. Slemrod,*Weak asymptotic decay via a “relaxed invariance principle” for a wave equation with nonlinear, nonmonotone damping*, Proc. Roy. Soc. Edinburgh Sect. A**113**(1989), no. 1-2, 87–97. MR**1025456**, 10.1017/S0308210500023970**[23]**V. A. Tupčiev,*The asymptotic behavior of the solution of the Cauchy problem for the equation 𝜖²𝑡𝑢ₓₓ=𝑢_{𝑡}+[𝜙(𝑢)]ₓ that degenerates for 𝜖=0 into the problem of the decay of an arbitrary lacuna for the case of a rarefaction wave*, Ž. Vyčisl. Mat. i Mat. Fiz.**12**(1972), 770–775 (Russian). MR**0306689****[24]**V. A. Tupčiev,*The method of introducing a viscosity in the study of a problem of decay of a discontinuity*, Dokl. Akad. Nauk SSSR**211**(1973), 55–58 (Russian). MR**0330801**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1992-1104200-7

Keywords:
Viscosity-capillarity travelling wave criterion

Article copyright:
© Copyright 1992
American Mathematical Society