arXiv 2103.122302021-03-22math.AP

光滑初值下一阶双曲型守恒律多维激波的形成与构造

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

Yin Huicheng, Zhu Lu

备注：: 24 pages, 5 figures

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摘要：

本文研究了具有光滑初值$u_0(x,y)$的一阶守恒律$\partial_t u+\partial_x F(u)+\partial_y G(u)=0$的多维激波的形成和构造问题。当$\min{H(ξ,η})<0$对$H(ξ,η)=\partial_ξ(F'(u_0(ξ,η)))+\partial_η(G'(u_0(ξ,η)))$成立时，光滑解$u$在时间$T^*=-\frac{1}{\min{H(ξ,η)}}$上爆破，更准确地说，只有一阶导数$\nabla_{t,x,y}u$在$t=T^*$上爆破，而$u$本身仍然连续，直到$t=T^*$ .在$H(ξ,η)$的一般非退化条件下，构造了在激波面$Σ$两侧非一致Lipschitz连续的$t\ge T^*$的局部弱熵解$u$.构造激波的强度在初始爆破曲线$Γ$上为零，然后在$t>T^*$上逐渐增加。此外，在$Γ$的邻域内，对解$u$的奇异性给出了详细而精确的描述。

英文摘要：

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 $T^*=-\frac{1}{\min{H(ξ,η)}}$ when $\min{H(ξ,η})<0$ holds for $H(ξ,η)=\partial_ξ(F'(u_0(ξ,η)))+\partial_η(G'(u_0(ξ,η)))$, 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 $H(ξ,η)$, 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 $Σ$. The strength of the constructed shock is zero on the initial blowup curve $Γ$ and then gradually increases for $t>T^*$. Additionally, in the neighbourhood of $Γ$, some detailed and precise descriptions on the singularities of solution $u$ are given.

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