Nonlinear Evolution Equations That Change Type pp 218-231 | Cite as

# Undercompressive Shocks in Systems of Conservation Laws

Conference paper

## Abstract

In this paper, we describe recent progress in our understanding of Riemann problems that involve undercompressive shock waves for 2 × 2 systems of nonstrictly hyperbolic conservation laws. A 2 × 2 system of conservation laws , , where

$$ {U_t} + F{(U)_x} = 0 $$

(1.1)

*U = U(x,t)*∈**R**^{2},*F*:**R**^{2}→**R**^{2}, is*nonstrictly hyperbolic*if the eigenvalues λ_{1}(*U*) ≤ λ_{2}(*U*) of*dF(U)*are real, but not distinct for every*U*. As defined in [4], system (1.1) has an*umbilic point*at*U = U**if*dF(U*)*is a multiple of the identity. Hyperbolic equations with an isolated umbilic point can be classified locally according to properties of the quadratic map*d*^{2}*F(U*)*. Since linear changes of coordinates do not affect the shocks or rarefaction waves for quadratic nonlinearities*F*, the general family of quadratic nonlinearities*F*with a unique umbilic point can be reduced to a two parameter family, which we write as$$ Q(u,v) = d(a{u^3}/3 + b{u^2}v + u{v^2}),a \ne 1 + {b^2} $$

(1.2)

*d*denotes gradient with respect to*U = (u,v)*.## Keywords

Bifurcation Diagram Riemann Problem Quadratic Nonlinearity Saddle Connection Melnikov Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## References

- [1]C.C. Chicone,
*Quadratic gradients on the plane are generically Morse-Smale*, J. Differential Equations, 33 (1979), pp. 159–161.MathSciNetMATHCrossRefGoogle Scholar - [2]M. Golubitsky and D.G. Schaeffer,
*Singularities and Groups in Bifurcation Theory*, Springer, New York, 1985.MATHGoogle Scholar - [3]
- [4]D.G. Schaeffer and M. Shearer,
*The classification of*2 × 2*systems of nonstrictly hyperbolic conservation laws, with application to oil recovery*, Comm. Pure Appl. Math., 40 (1987), pp. 141–178.MathSciNetMATHCrossRefGoogle Scholar - [5]D.G. Schaeffer and M. Shearer,
*Riemann problems for nonstrictly hyperbolic*2 × 2*systems of conservation laws*, Trans. Amer. Math. Soc., 304 (1987), pp. 267–306.MathSciNetMATHGoogle Scholar - [6]S. Schecter and M. Shearer,
*Undercompressive shocks for nonstrictly hyperbolic conservation laws*, IMA preprint.Google Scholar - [7]M. Shearer, D.G. Schaeffer, D. Marchesin and P.J. Paes-Leme,
*Solution of the Riemann problem for a prototype*2 × 2*system of nonstrictly hyperbolic conservation laws*, Arch. Rat. Mech. Anal., 97 (1987), pp. 299–320.MathSciNetMATHCrossRefGoogle Scholar - [8]M. Shearer,
*The Riemann problem for*2 × 2*systems of hyperbolic conservation laws with Case I quadratic nonlinearities*, J. Differential Equations, 80 (1989), pp. 343–363.MathSciNetMATHCrossRefGoogle Scholar

## Copyright information

© Springer-Verlag New York Inc. 1990