Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



A nonconvex scalar conservation law with trilinear flux

Authors: Brian T. Hayes and Michael Shearer
Journal: Quart. Appl. Math. 59 (2001), 615-635
MSC: Primary 35L65; Secondary 35B25, 35L60
DOI: https://doi.org/10.1090/qam/1866551
MathSciNet review: MR1866551
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Abstract | References | Similar Articles | Additional Information

Abstract: The focus of this paper is on traveling wave solutions of the equation

$\displaystyle {u_t} + f{\left( u \right)_x} = \epsilon {u_{xx}} + {\epsilon ^2}\gamma {u_{xxx}}$

, in which the flux function $ f$ is trilinear and nonconvex. In particular, it is shown that for combinations of parameters in certain ranges, there are traveling waves that converge as $ \epsilon \to 0$ to undercompressive shocks, in which the characteristics pass through the shock. The analysis is based on explicit solutions of the piecewise linear ordinary differential equation satisfied by traveling waves. The analytical results are illustrated by numerical solutions of the Riemann initial value problem, and are compared with corresponding explicit results for the case of a cubic flux function.

References [Enhancements On Off] (What's this?)

  • [1] R. Abeyaratne and J. K. Knowles, Implications of viscosity and strain gradient effects for the kinetics of propagating phase boundaries in solids, SIAM J. Appl. Math. 51, 1205-1221 (1991) MR 1127848
  • [2] B. T. Hayes and P. G. LeFloch, Nonclassical shock waves: Scalar conservation laws, Arch. Rat. Mech. Anal. 139, 1-56 (1997) MR 1475777
  • [3] B. T. Hayes and P. G. LeFloch, Nonclassical shock waves and kinetic relations: Finite difference schemes, SIAM J. Numer. Anal. 35, 2169-2194 (1998) MR 1655842
  • [4] B. T. Hayes and M. Shearer, Undercompressive shocks and Riemann problems for scalar conservation laws with non-convex fluxes, Proc. Royal Soc. Edinburgh Sect. A 129, 733-754 (1999) MR 1718538
  • [5] E. Isaacson, D. Marchesin, and B. Plohr, Transitional waves for conservation laws, SIAM J. Math. Anal. 21, 837-866 (1990) MR 1052875
  • [6] D. Jacobs, W. McKinney, and M. Shearer, Travelling wave solutions of the modified Korteweg-de Vries-Burgers equation, J. Differential Equations 116, 448-467 (1995) MR 1318583
  • [7] P. D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10, 537-566 (1957) MR 0093653
  • [8] A. Azevedo, D. Marchesin, B. Plohr, and K. Zumbrun, Bifurcation of nonclassical viscous shock profiles from the constant state, Comm. Math. Phys. 202, 267-290 (1999) MR 1689983
  • [9] M. Schulze and M. Shearer, Undercompressive shocks for a system of hyperbolic conservation laws with cubic nonlinearity, J. Math. Anal. Appl. 229, 344-362 (1999) MR 1664281
  • [10] M. Shearer and Y. Yang, The Riemann problem for a system of conservation laws of mixed type with a cubic nonlinearity, Proc. Royal Soc. Edinburgh Sect. A 125, 675-699 (1995) MR 1357378
  • [11] M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Arch. Rat. Mech. Anal. 81, 301-315 (1983) MR 683192
  • [12] C. C. Wu, New theory of MHD shock waves, Viscous Profiles and Numerical Methods for Shock Waves (ed. M. Shearer), SIAM, Philadelphia, PA, 1991, pp. 209-236 MR 1142652

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DOI: https://doi.org/10.1090/qam/1866551
Article copyright: © Copyright 2001 American Mathematical Society

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