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

TL;DR

This paper investigates the formation of multidimensional shock waves in first-order hyperbolic conservation laws with smooth initial data, constructing weak solutions beyond blowup time and analyzing shock properties and singularities.

Contribution

It introduces a method to construct local weak entropy solutions after shock formation for multidimensional conservation laws with smooth initial data.

Findings

Shock strength is initially zero and increases over time.

Constructed solutions are not uniformly Lipschitz continuous across the shock surface.

Detailed descriptions of solution singularities near the initial blowup curve.

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.