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# 3-D Kinematical Conservation Laws (KCL): Equations of Evolution of a Surface

Arun, KR and Prasad, Phoolan (2007) 3-D Kinematical Conservation Laws (KCL): Equations of Evolution of a Surface. [Preprint]

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## Abstract

We discuss various aspects of the KCL of Giles, Prasad and Ravindran (GPR) (1995) in 3-space dimensions. These are equations of evolution of a moving surface $\Omega_t$ and are derived on the assumption that the motion of the surface is isotropic so that we can associate a simpler form of the ray velocity $\mbox{\boldmath{$\chi$}}=m \mathbf{n}$ at a point of $\Omega_t$, where ${\bf n}$ is unit normal of $\Omega_t$ and $m$ is a scalar independent of ${\bf n}$. These equations are the most general equations in conservation form, governing the evolution of a {\bf propagating} surface $\Omega_t$ with singularities which we call {\bf kinks} and which are curves across which the normal $\mathbf{n}$ to $\Omega_t$ and amplitude $w$ on $\Omega_t$ are discontinuous. Following GPR, we discuss the jump relations across a kink and some of their properties. The amplitude $w$ of $\Omega_t$ is related to the normal velocity $m$ of $\Omega_t$ and this relation is to be determined from the physical laws governing the evolution of $\Omega_t$. We derive a simpler system of six differential equations from KCL and show that the KCL system is equivalent to the ray equations for $\Omega_t$. The six independent equations and an energy transport equation involving $w$ (in a simplified model) form a completely determined system of seven equations, which has a pair of nonzero eigenvalues and five zero eigenvalues. The dimension of the eigenspace associated with the multiple eigenvalue $0$ is $4$ i.e., this eigenspace is not complete. For an appropriately defined normal velocity, the two nonzero eigenvalues are real when $m>1$ and pure imaginary when $m<1$. Though determination of the eigenvalues is extremely complicated, our results on eigenvalues and eigenvectors have been obtained indirectly by a transformation and are also supported by numerical solution of the 7th degree characteristic equation. We shall present the numerical solutions of examples of propagating surfaces using this theory in subsequent papers.

Item Type: Preprint http://www.math.iisc.ernet.in/~prasad/pr... Ray theory;kinematical conservation laws;nonlinear waves;conservation laws;shock propagation;curved shock;hyperbolic and elliptic systems;Fermat's principle Division of Physical & Mathematical Sciences > Mathematics 11 Jan 2008 27 Aug 2008 13:09 http://eprints.iisc.ernet.in/id/eprint/12887

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