Multiple viscous solutions for systems of conservation laws
Authors:
A. V. Azevedo and D. Marchesin
Journal:
Trans. Amer. Math. Soc. 347 (1995), 30613077
MSC:
Primary 35L65; Secondary 35M10, 76L05
MathSciNet review:
1277093
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Abstract: We exhibit an example of mechanism responsible for multiple solutions in the Riemann problem for a mixed elliptichyperbolic type system of two quadratic polynomial conservation laws. In this example, multiple solutions result from folds in the set of Riemann solutions. The multiple solutions occur despite the fact that they all satisfy the viscous profile entropy criterion. The failure of this criterion to provide uniqueness is evidence in support of a need for conceptual change in the theory of shock waves for a system of conservation laws.
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 [1]
 A. A. Andronov, A. A. Vitt, and S. E. Khaikin, Theory of oscillations, AddisonWesley, Reading, MA, 1966.
 [2]
 A. V. Azevedo and D. Marchesin, Multiple viscous profile Riemann solutions in mixed elliptichyperbolic models for flow in porous media, IMA Vol. Math. Appl., vol 27, SpringerVerlag, New York and Berlin, 1990, pp. 117. MR 1074181 (91f:35183)
 [3]
 A. V. Azevedo, Multiple fundamental solutions in elliptichyperbolic systems of conservation laws, Ph. D. Thesis (in Portuguese), Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, Brazil, 1991.
 [4]
 J. Bell, J. Trangenstein, and G. Shubin, Conservation laws of mixed type describing threephase flow in porous media, SIAM J. Appl. Math. 46 (1986), 10001017. MR 866277 (87m:76058)
 [5]
 S. Canic and B. Plohr, Shock wave admissibility for quadratic conservation laws, Proc. XVII Colóquio Brasileiro de Matemática, IMPA, 1991, pp. 199216.
 [6]
 C. Chicone, Quadratic gradients on the plane are generically MorseSmale, J. Differential Equations 33 (1979), 159166. MR 542667 (80g:58036)
 [7]
 J. Glimm and D. Sharp, An matrix theory for classical nonlinear physics, Found. Phys. 16 (1986), 125141. MR 836850 (88c:35141)
 [8]
 C. C. Conley and J. A. Smoller, Viscosity matrices for twodimensional nonlinear hyperbolic systems, Comm. Pure Appl. Mat. 22 (1970), 867884. MR 0274956 (43:714)
 [9]
 R. Courant and K. O. Friedrichs, Supersonic flow and shock waves, Interscience, New York, 1948. MR 0029615 (10:637c)
 [10]
 F. J. Fayers and J. D. Matthews, Evaluation of normalized Stone's methods for estimating threephase relative permeabilities, Soc. Petrol. Engin. J. 24 (1984), 225232.
 [11]
 I. M. Gel'fand, Some problems in theory of quasilinear equations, Uspekhi Mat. Nauk 14 (1959), 87158. MR 0110868 (22:1736)
 [12]
 , Some problems in theory of quasilinear equations, Amer. Math. Soc. Transl. Ser. 2, vol. 29, Amer. Math. Soc., Providence, RI, 1963, pp. 295381. MR 0153960 (27:3921)
 [13]
 H. Holden, On the Riemann problem for a prototype of a mixed type conservation law, Comm. Pure Appl. Math. 40 (1987), 229264. MR 872386 (88d:35125)
 [14]
 E. Isaacson, D. Marchesin, and B. Plohr, Transitional waves for conservation laws, SIAM J. Math. Anal. 21 (1990), 837866. MR 1052875 (91e:35129)
 [15]
 E. Isaacson, D. Marchesin, B. Plohr, and B. Temple, The classification of solutions of quadratic Riemann problems I, SIAM J. Appl. Math. 48 (1988), 10091032. MR 960467 (89k:35139)
 [16]
 E. Isaacson and J. B. Temple, The structure of asymptotic states in a singular system of conservation laws, Adv. Appl. Math. 11 (1990), 205219. MR 1053229 (91h:35205)
 [17]
 P. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 19 (1957), 537566. MR 0093653 (20:176)
 [18]
 D. Marchesin and P. J. PaesLeme, A Riemann problem with bifurcation in gas dynamics, J. Comput. Math. Appl. 12 A (1986), 433455. MR 841979 (87e:76106)
 [19]
 C. F. Palmeira, Line fields defined by eigenspaces of derivatives of maps from the plane to itself, Proc. VI International Conference of Differential Geometry, Univ. Santiago de Compostela, Spain, 1988, pp. 177205. MR 1040846 (91a:58099)
 [20]
 O. A. Oleinik, On the uniqueness of generalized solution of Cauchy problem for non linear system of equations occurring in mechanics, Uspekhi Mat. Nauk (Russian Math. Surveys) 12 (1957), 169176. MR 0094543 (20:1057)
 [21]
 D. Shaeffer, M. Shearer, D. Marchesin, and P. PaesLeme, Solution of Riemann problem for a prototype system of nonstrictly hyperbolic conservation laws, Arch. Rational Mech. Anal. 97 (1987), 299320. MR 865843 (88a:35156)
 [22]
 H. Whitney, On singularities of mappings Euclidean spaces: Imapping from the plane to the plane, Ann. of Math. (2) 62 (1955), 347410. MR 0073980 (17:518d)
 [23]
 Ye YanQian et al., Theory of limit cycles, Transl. Math. Monographs, Amer. Math. Soc., Providence, RI, 1984.
 [24]
 K. Zumbrum, B. Plohr, and D. Marchesin, Scattering behavior of transitional shock waves, Mat. Contemp. 3 (1992), 191209. MR 1303177 (95f:35153)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199512770938
PII:
S 00029947(1995)12770938
Article copyright:
© Copyright 1995
American Mathematical Society
