Seminar

In a large number of physical phenomena, we find propagating surfaces which need mathematical treatment. d-D kinematical conservation laws (KCL) are equations of evolution of a moving surface Ω_t in d-dimensional (x₁ , x₂ , . . . , x_d )-space R^d . The KCL are derived in a specially defined ray coordinates (ξ₁ , ξ₂ , . . . , ξ_d−1 , t), where ξ₁ , ξ₂ , . . . , ξ_d−1 are surface coordinates on Ω_t and t is time. KCL are the most general equations in conservation form, governing the evolution of Ω_t with physically realistic singularities. A very special type of singularity is a kink, which is a point on Ω_t when Ω_t is a curve in R² and is a curve on Ω_t when it is a surface in R³ . Across a kink the normal n to Ω_t and normal velocity m on Ω_t are discontinuous. Since the KCL system contains only kinematical relations, it is an under-determined system of equations. In order to complete the system, we need to find additional equations representing the dynamics of Ω_t from the governing equations of the medium in which Ω_t propagates. The mathematical analysis of 3-D KCL system and computation with its help present a challenge since the eigenspace is not complete and there are geometric solenoidal constrains. We present a few examples of Ω_t and numerical results.