Fronts Propagating with Curvature- Dependent Speed: Algorithms ...

ALGORITHMS FOR SURFACE MOTION PROBLEMS 17
mimic Oleinik’s proof only-the more general BV case follows as in [17]. Let
U(s, t) = 1 if O,(S, t) > 0, - 1 if 6Js, t) < 0, and 0 if 6,(s, t) = 0. Then,
Let [si, si+,] be an interval on which 8,>0 with 8, vanishing at the end
Then
---- Hds
=F, g 0s e/, gs si+i 1 SF’ !% !$“(‘.
0
4 0 g g s,
The first term on the right vanishes because 6, = 0 at each end point; the second
term is non-positive because -F’(O) z 0 and QSs(s,+ I) d 0 < B,,(s~). A similar
argument works on intervals for which 8, < 0 in the interior and 8, vanishes at the
end points. This completes the proof.
Using Eqs. (2.3) and (2.4), we have
K,= -[;F(K)/g]s;-K’F(K).
Letting P= 1 - EK, we obtain, as in [32],
and, for e = 0,
K, = &Kss + EK’ - K2, (2.5)
W 0)
K(s’ ‘) = (1 + tK(s, 0))’
This becomes infinite in finite time if K(s, 0) is anywhere negative and is analogous
to shock formation experienced in the single scalar convex conservation law. ore
precisely, consider the “viscous” conservation law with G concave, namely
If we take E = 0 (the shock case), weak solutions to
u, + CG(u)l, = 0
4x, 0) = uob)