The shock formation and optimal regularities of the resulting shock curves for 1D scalar conservation laws*

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