Speaker

:

Prof. Phoolan Prasad

Subject Area

:

Mathematics

Venue

:

Lecture Hall - III, Dept of Mathematics

Time

:

4.00 pm

Date  

:

February 22,2008 (Friday)

Title

:

3-D Kinematical Conservation Laws: equations of evolution of a surface

We discuss various proprties of the KCL of Giles, Prasad and Ravindran (GPR) (1995) in 3-space dimensions. These equations
of evolution of a {\bf propagating} surface $\Omega_t$, are derived assuming the motion of the surface to be isotropic. These are the most general equations in conservation form, governing the evolution of $\Omega_t$ 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. 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 $w$ (which is related to the normal velocity $m$ of $\Omega_t$) form a completely determined system of seven equations. We have determined eigenvalues of the system by a very novel method and find that the system has two distinct
nonzero eigenvalues and five zero eigenvalues and the dimension of the eigenspace associated with the multiple eigenvalue $0$ is only $4$. For an appropriately defined $m$, the two nonzero eigenvalues are real when $m>1$ and pure imaginary when $m<1$. Finally we have shown how to use the theory to concrete examples.