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 ${\partial }_{t}u+{\partial }_{x}f(u)=0$ with the smooth initial data $u(0,x)=u_0(x)$. If $u_0(x)\in C^1(\mathbb{R})$ and $f(u)\in C^2(\mathbb{R})$, it is well-known that the solution $u$ will blow up on the time $T^{\ast }=-\frac{1}{\min g^{\prime }(x)}$ when $\min g^{\prime }(x)<0$ holds for $g(x)=f^{\prime }(u_0(x))$. Let $x_0$ be a local minimum point of $g^{\prime }(x)$ such that $g^{\prime }(x_0)=\min g^{\prime }(x)<0$ and $g^{\prime \prime }(x_0)=0$, $g^{(3)}(x_0)>0$ (which is called the generic nondegenerate condition), then by Theorem 2 of \cite{Le94}, a weak entropy solution $u$ together with the shock curve $x=\phi (t)\in C^2[T^{\ast },T^{\ast }+\epsilon )$ starting from the blowup point $(T^{\ast },x^{\ast }=x_0+g(x_0)T^{\ast })$ can be locally constructed. When the generic nondegenerate condition is violated, namely, when $x_0$ is a local minimum point of $g^{\prime }(x)$ such that $g^{\prime \prime }(x_0)=g^{(3)}(x_0)=...=g^{(2k_0)}(x_0)=0$ but $g^{(2k_0+1)}(x_0)>0$ for some $k_0\in \mathbb{N}$ with $k_0\ge 2$; or $g^{(k)}(x_0)=0$ for any $k\in \mathbb{N}$ and $k\ge 2$, we will study the shock formation and the optimal regularity of the shock curve $x=\phi (t)$, meanwhile, some precise descriptions on the behaviors of $u$ near the blowup point $(T^{\ast },x^{\ast })$ 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.

arXiv comment31 pages, 2 figures

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