Speaker :
| Professor Henrik Kalisch (University of Bergen) |

Title :
| On the Existence of Singular Solutions for Systems of Conservation Laws |

Time :
| 2017-05-05 (Fri) 11:00 - 12:00 |

Place : | Seminar Room 617, Institute of Mathematics (NTU Campus) |

Abstract: |
Existence and admissibility of delta-shock solutions is discussed for
hyperbolic systems of conservation laws, with a focus on systems which
do not admit classical Lax-admissible solution to certain Riemann
problems. By introducing complex-valued corrections in the framework
of the weak asymptotic method, we show that a compressive delta-shock
wave solution resolves such Riemann problems. By letting the
approximation parameter tend to zero, the corrections become real
valued and the resulting distributions fit into a generalized concept
of singular solutions [V. G. Danilov and V. M. Shelkovich, Dynamics of
propagation and interaction of delta-shock waves in hyperbolic
systems, J. Differential Equations 211 (2005), 333-381]. In this
framework, it can be shown that every 2x2 system of conservation laws
admits delta-shock solutions.
As an example, singular solution of the classical shallow-water equations are investigated. It is shown that the combination of discontinuous free-surface solutions and bottom step transitions naturally leads to singular solutions featuring Dirac delta distributions. |