Multiple viscous solutions for systems of conservation laws

Authors:
A. V. Azevedo and D. Marchesin

Journal:
Trans. Amer. Math. Soc. **347** (1995), 3061-3077

MSC:
Primary 35L65; Secondary 35M10, 76L05

MathSciNet review:
1277093

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[1]**A. A. Andronov, A. A. Vitt, and S. E. Khaikin,*Theory of oscillations*, Addison-Wesley, Reading, MA, 1966.**[2]**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**, 10.1007/978-1-4613-9049-7_1**[3]**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.**[4]**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**, 10.1137/0146059**[5]**S. Canic and B. Plohr,*Shock wave admissibility for quadratic conservation laws*, Proc. XVII Colóquio Brasileiro de Matemática, IMPA, 1991, pp. 199-216.**[6]**Carmen C. Chicone,*Quadratic gradients on the plane are generically Morse-Smale*, J. Differential Equations**33**(1979), no. 2, 159–166. MR**542667**, 10.1016/0022-0396(79)90085-8**[7]**J. Glimm and D. H. Sharp,*An 𝑆 matrix theory for classical nonlinear physics*, Found. Phys.**16**(1986), no. 2, 125–141. MR**836850**, 10.1007/BF01889377**[8]**Charles C. Conley and Joel A. Smoller,*Viscosity matrices for two-dimensional nonlinear hyperbolic systems.*, Comm. Pure Appl. Math.**23**(1970), 867–884. MR**0274956****[9]**R. Courant and K. O. Friedrichs,*Supersonic Flow and Shock Waves*, Interscience Publishers, Inc., New York, N. Y., 1948. MR**0029615****[10]**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.**[11]**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****[12]**I. M. Gel′fand,*Some problems in the theory of quasilinear equations*, Amer. Math. Soc. Transl. (2)**29**(1963), 295–381. MR**0153960****[13]**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**, 10.1002/cpa.3160400206**[14]**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**, 10.1137/0521047**[15]**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**, 10.1137/0148059**[16]**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**, 10.1016/0196-8858(90)90009-N**[17]**P. D. Lax,*Hyperbolic systems of conservation laws. II*, Comm. Pure Appl. Math.**10**(1957), 537–566. MR**0093653****[18]**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****[19]**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****[20]**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****[21]**M. Shearer, D. G. Schaeffer, D. Marchesin, and P. L. Paes-Leme,*Solution of the Riemann problem for a prototype 2×2 system of nonstrictly hyperbolic conservation laws*, Arch. Rational Mech. Anal.**97**(1987), no. 4, 299–320. MR**865843**, 10.1007/BF00280409**[22]**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**0073980****[23]**Ye Yan-Qian et al.,*Theory of limit cycles*, Transl. Math. Monographs, Amer. Math. Soc., Providence, RI, 1984.**[24]**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**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
35L65,
35M10,
76L05

Retrieve articles in all journals with MSC: 35L65, 35M10, 76L05

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1995-1277093-8

Article copyright:
© Copyright 1995
American Mathematical Society