Multiple viscous solutions for systems of conservation laws
HTML articles powered by AMS MathViewer
- by A. V. Azevedo and D. Marchesin PDF
- Trans. Amer. Math. Soc. 347 (1995), 3061-3077 Request permission
Abstract:
We exhibit an example of mechanism responsible for multiple solutions in the Riemann problem for a mixed elliptic-hyperbolic 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.References
-
A. A. Andronov, A. A. Vitt, and S. E. Khaikin, Theory of oscillations, Addison-Wesley, Reading, MA, 1966.
- A. V. Azevedo and D. Marchesin, Multiple viscous profile Riemann solutions in mixed elliptic-hyperbolic models for flow in porous media, Nonlinear evolution equations that change type, IMA Vol. Math. Appl., vol. 27, Springer, New York, 1990, pp. 1–17. MR 1074181, DOI 10.1007/978-1-4613-9049-7_{1} A. V. Azevedo, Multiple fundamental solutions in elliptic-hyperbolic systems of conservation laws, Ph. D. Thesis (in Portuguese), Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, Brazil, 1991.
- John B. Bell, John A. Trangenstein, and Gregory R. Shubin, Conservation laws of mixed type describing three-phase flow in porous media, SIAM J. Appl. Math. 46 (1986), no. 6, 1000–1017. MR 866277, DOI 10.1137/0146059 S. Canic and B. Plohr, Shock wave admissibility for quadratic conservation laws, Proc. XVII Colóquio Brasileiro de Matemática, IMPA, 1991, pp. 199-216.
- Carmen C. Chicone, Quadratic gradients on the plane are generically Morse-Smale, J. Differential Equations 33 (1979), no. 2, 159–166. MR 542667, DOI 10.1016/0022-0396(79)90085-8
- J. Glimm and D. H. Sharp, An $S$ matrix theory for classical nonlinear physics, Found. Phys. 16 (1986), no. 2, 125–141. MR 836850, DOI 10.1007/BF01889377
- Charles C. Conley and Joel A. Smoller, Viscosity matrices for two-dimensional nonlinear hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 867–884. MR 274956, DOI 10.1002/cpa.3160230603
- R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, N. Y., 1948. MR 0029615 F. J. Fayers and J. D. Matthews, Evaluation of normalized Stone’s methods for estimating three-phase relative permeabilities, Soc. Petrol. Engin. J. 24 (1984), 225-232.
- I. M. Gel′fand, Some problems in the theory of quasi-linear equations, Uspehi Mat. Nauk 14 (1959), no. 2 (86), 87–158 (Russian). MR 0110868
- I. M. Gel′fand, Some problems in the theory of quasilinear equations, Amer. Math. Soc. Transl. (2) 29 (1963), 295–381. MR 0153960, DOI 10.1090/trans2/029/12
- Helge Holden, On the Riemann problem for a prototype of a mixed type conservation law, Comm. Pure Appl. Math. 40 (1987), no. 2, 229–264. MR 872386, DOI 10.1002/cpa.3160400206
- Eli L. Isaacson, Dan Marchesin, and Bradley J. Plohr, Transitional waves for conservation laws, SIAM J. Math. Anal. 21 (1990), no. 4, 837–866. MR 1052875, DOI 10.1137/0521047
- E. Isaacson, D. Marchesin, B. Plohr, and B. Temple, The Riemann problem near a hyperbolic singularity: the classification of solutions of quadratic Riemann problems. I, SIAM J. Appl. Math. 48 (1988), no. 5, 1009–1032. MR 960467, DOI 10.1137/0148059
- Eli Isaacson and Blake Temple, The structure of asymptotic states in a singular system of conservation laws, Adv. in Appl. Math. 11 (1990), no. 2, 205–219. MR 1053229, DOI 10.1016/0196-8858(90)90009-N
- P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 93653, DOI 10.1002/cpa.3160100406
- D. Marchesin and P. J. Paes-Leme, A Riemann problem in gas dynamics with bifurcation, Comput. Math. Appl. Part A 12 (1986), no. 4-5, 433–455. Hyperbolic partial differential equations, III. MR 841979
- C. F. B. Palmeira, Line fields defined by eigenspaces of derivatives of maps from the plane to itself, Proceedings of the Sixth International Colloquium on Differential Geometry (Santiago de Compostela, 1988) Cursos Congr. Univ. Santiago de Compostela, vol. 61, Univ. Santiago de Compostela, Santiago de Compostela, 1989, pp. 177–205. MR 1040846
- O. A. Oleĭnik, On the uniqueness of the generalized solution of the Cauchy problem for a non-linear system of equations occurring in mechanics, Uspehi Mat. Nauk (N.S.) 12 (1957), no. 6(78), 169–176 (Russian). MR 0094543
- M. Shearer, D. G. Schaeffer, D. Marchesin, and P. L. Paes-Leme, Solution of the Riemann problem for a prototype $2\times 2$ system of nonstrictly hyperbolic conservation laws, Arch. Rational Mech. Anal. 97 (1987), no. 4, 299–320. MR 865843, DOI 10.1007/BF00280409
- Hassler Whitney, On singularities of mappings of euclidean spaces. I. Mappings of the plane into the plane, Ann. of Math. (2) 62 (1955), 374–410. MR 73980, DOI 10.2307/1970070 Ye Yan-Qian et al., Theory of limit cycles, Transl. Math. Monographs, Amer. Math. Soc., Providence, RI, 1984.
- Kevin R. Zumbrun, Bradley J. Plohr, and Dan Marchesin, Scattering behavior of transitional shock waves, Mat. Contemp. 3 (1992), 191–209. Second Workshop on Partial Differential Equations (Rio de Janeiro, 1991). MR 1303177
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 347 (1995), 3061-3077
- MSC: Primary 35L65; Secondary 35M10, 76L05
- DOI: https://doi.org/10.1090/S0002-9947-1995-1277093-8
- MathSciNet review: 1277093