论文标题
一维$ n \ times n $严格的双曲线保护法的形成和构建和构建具有较小平滑初始数据
Formation and construction of a shock wave for one dimensional $n\times n$ strictly hyperbolic conservation laws with small smooth initial data
论文作者
论文摘要
根据1-D $ n \ times n $严格双曲线保护法的真正非线性假设，我们研究了平滑溶液的几何爆炸以及小的初始数据满足一般的非排定条件时的奇异性。首先，在唯一的爆炸点附近，我们对光滑解决方案的时空爆炸速率进行了精确描述，同时得出了特征封底的尖尖象奇异性结构。通过在整个爆炸时间内扩展完全非线性爆炸系统的平滑解决方案来确定这些结果。随后，通过使用$（n-1）$好的组件和一个不良组件的生成的1-D严格双曲系统上的新形式，以及选择有效的迭代方案以及一些涉及的分析，构建了从爆炸点开始的弱熵冲击波。作为副产品，我们的结果可以应用于2-D超音速稳定可压缩全欧式方程（$ 4 \ times 4 $系统），1-D MHD方程（$ 5 \ times 5 $系统），1-D弹性波方程（$ 6 \ tims 6 $ 6 $ 6 $ 6 $ 6 $ 6 $ 6 $ 6 $ 6 $ 6 $ $ 7 $ 7 \ $ 7 \ $ 7 \ $ 7 \ $ 7 \ $ 7 \ $ 7 \ $ 7 \ $ 7 \ $ 7）。
Under the genuinely nonlinear assumption for 1-D $n\times n$ strictly hyperbolic conservation laws, we investigate the geometric blowup of smooth solutions and the development of singularities when the small initial data fulfill the generic nondegenerate condition. At first, near the unique blowup point we give a precise description on the space-time blowup rate of the smooth solution and meanwhile derive the cusp singularity structure of characteristic envelope. These results are established through extending the smooth solution of the completely nonlinear blowup system across the blowup time. Subsequently, by utilizing a new form on the resulting 1-D strictly hyperbolic system with $(n-1)$ good components and one bad component, together with the choice of an efficient iterative scheme and some involved analyses, a weak entropy shock wave starting from the blowup point is constructed. As a byproduct, our result can be applied to the shock formation and construction for the 2-D supersonic steady compressible full Euler equations ($4\times 4$ system), 1-D MHD equations ($5\times 5$ system), 1-D elastic wave equations ($6\times 6$ system) and 1-D full ideal compressible MHD equations ($7\times 7$ system).
