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.