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 $T^*=-\frac{1}{\min{H(\xi,\eta)}}$ when $\min{H(\xi,\eta})<0$ holds for $H(\xi,\eta)=\partial_{\xi}(F'(u_0(\xi,\eta)))+\partial_{\eta}(G'(u_0(\xi,\eta)))$, more precisely, only the first order derivatives $\nabla_{t,x,y}u$ blow up on $t=T^*$ meanwhile $u$ itself is still continuous until $t=T^*$. Under the generic nondegenerate condition of $H(\xi,\eta)$, we construct a local weak entropy solution $u$ for $t\ge T^*$ 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 $t>T^*$. Additionally, in the neighbourhood of $\Gamma$, some detailed and precise descriptions on the singularities of solution $u$ are given.