  The study on the shock formation and the regularities of the resulting shock
surfaces for hyperbolic conservation laws is a basic problem in the nonlinear
partial differential equations. In this paper, we are concerned with the shock
formation and the optimal regularities of the resulting shock curves for the
1-D conservation law $\partial_tu+\partial_xf(u)=0$ with the smooth initial
data $u(0,x)=u_0(x)$. If $u_0(x)\in C^{1}(\Bbb R)$ and $f(u)\in C^2(\Bbb R)$,
it is well-known that the solution $u$ will blow up on the time
$T^*=-\frac{1}{\min{g'(x)}}$ when $\min{g'(x)}<0$ holds for $g(x)=f'(u_0(x))$.
Let $x_0$ be a local minimum point of $g'(x)$ such that $g'(x_0)=\min{g'(x)}<0$
and $g''(x_0)=0$, $g^{(3)}(x_0)>0$ (which is called the generic nondegenerate
condition), then by Theorem 2 of \cite{Le94}, a weak entropy solution $u$
together with the shock curve $x=\varphi(t)\in C^2[T^*, T^*+\varepsilon)$
starting from the blowup point $(T^*, x^*=x_0+g(x_0)T^*)$ can be locally
constructed. When the generic nondegenerate condition is violated, namely, when
$x_0$ is a local minimum point of $g'(x)$ such that
$g''(x_0)=g^{(3)}(x_0)=...=g^{(2k_0)}(x_0)=0$ but $g^{(2k_0+1)}(x_0)>0$ for
some $k_0\in\Bbb N$ with $k_0\ge 2$; or $g^{(k)}(x_0)=0$ for any $k\in\Bbb N$
and $k\ge 2$, we will study the shock formation and the optimal regularity of
the shock curve $x=\varphi(t)$, meanwhile, some precise descriptions on the
behaviors of $u$ near the blowup point $(T^*, x^*)$ are given. Our main aims
are to show that: around the blowup point, the shock really appears whether the
initial data are degenerate with finite orders or with infinite orders; the
optimal regularities of the shock solution and the resulting shock curve have
the explicit relations with the degenerate degrees of the initial data.
