The Geometry of 2 × 2 Systems of Conservation Laws
- First Online:
We consider one typical two-parameter family of quadratic systems of 2 × 2 conservation laws, and study the geometry of the behaviour of the possible solutions of the Riemann problem near an umbilic point, following the geometric approach presented by Isaacson, Marchesin, Palmeira, Plohr, in A global formalism for nonlinear waves in conservation laws, Commun. Math. Phys. (1992). The corresponding phase portraits for the rarefaction curves, shock curves and composite curves are discussed.
Unable to display preview. Download preview PDF.
- 1.Arnold, V.: Singularities of Differentiable Maps, Vol. I, Birkhäuser, 1985.Google Scholar
- 2.Basto-Gonçalves, J. and Reis, H.: The Geometry of Quadratic 2 × 2 Systems of Conservation Laws, preprint 339, Université de Bourgogne, 2003.Google Scholar
- 6.Davydov, A.: Qualitative Theory of Control Systems, AMS Transl. Math. Monogr. 14, 1994.Google Scholar
- 8.Eschenazi, C. and Palmeira, C.: The structure of composite rarefaction-shock foliations for quadratic systems of conservation laws, Mat. Contemp. 22 (2002).Google Scholar
- 14.Palmeira, C.: Line fields defined by eigenspaces of derivatives of maps from the plane to itself, Proceedings of the IVth Conference of Differential Geometry, Santiago de Compostela (Spain), 1988.Google Scholar
- 15.Poston, T. and Stewart, I.: Catastrophe Theory and its Applications, Pitman, 1978.Google Scholar