Formation and construction of a multidimensional shock wave for the first order hyperbolic conservation law with smooth initial data

Abstract

In this paper, the problem on formation and construction of a multidimensional shock wave is studied for the first order conservation law ${\partial }_{t}u+{\partial }_{x}F(u)+{\partial }_{y}G(u)=0$ with smooth initial data $u_0(x,y)$. It is well-known that the smooth solution $u$ will blow up on the time $T^{\ast }=-\frac{1}{\min H(\xi ,\eta )}$ when $\min H(\xi ,\eta )<0$ holds for $H(\xi ,\eta )={\partial }_{\xi }(F^{\prime }(u_0(\xi ,\eta )))+{\partial }_{\eta }(G^{\prime }(u_0(\xi ,\eta )))$, more precisely, only the first order derivatives ${\nabla }_{t,x,y}u$ blow up on $t=T^{\ast }$ meanwhile $u$ itself is still continuous until $t=T^{\ast }$. Under the generic nondegenerate condition of $H(\xi ,\eta )$, we construct a local weak entropy solution $u$ for $t\ge T^{\ast }$ which is not uniformly Lipschitz continuous on two sides of a shock surface $\Sigma$. The strength of the constructed shock is zero on the initial blowup curve $\Gamma$ and then gradually increases for $t>T^{\ast }$. Additionally, in the neighbourhood of $\Gamma$, some detailed and precise descriptions on the singularities of solution $u$ are given.

24 pages, 5 figures