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

TL;DR

This paper investigates shock formation and regularity of shock curves in 1-D scalar conservation laws, especially under degenerate initial data conditions, providing precise descriptions and regularity relations.

Contribution

It extends shock formation analysis to degenerate cases where initial data conditions are of higher or infinite order, detailing shock regularities and behaviors.

Findings

Shock appears regardless of degeneracy degree in initial data.

Explicit relations between shock regularity and initial data degeneracy.

Precise behavior descriptions of solutions near blowup points.

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.