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

Abstract

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.