This article concerns the formation of finite-time singularities in solutions to quasilinear hyperbolic systems with small initial data. We propose a universal test function method that works for many nonlinear hyperbolic systems arising from physical applications. We first present a simpler proof of the main result in the work of Sideris [Commun. Math. Phys. 101(4), 475–485 (1985)]: the global classical solution is non-existent for compressible Euler equations even for some small initial data. Then, we apply this approach to nonlinear magnetohydrodynamics in two space dimensions. Finally, we consider second order quasilinear hyperbolic systems with quadratic nonlinearity arising from elastodynamics of isotropic hyperelastic materials by ignoring the cubic and higher order terms. Under some restriction on the coefficients of the nonlinear terms that imply genuine nonlinearity, we show that the classical solutions to these equations can still blow up in finite time even if the initial data are small enough.

中文翻译：

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具有小初始扰动的非线性双曲系统的有限时间奇点的形成

本文涉及在具有小初始数据的拟线性双曲系统的解中有限时间奇点的形成。我们提出了一种通用的测试函数方法，该方法适用于物理应用产生的许多非线性双曲线系统。我们首先在 Sideris [Commun. 数学。物理。101(4), 475–485 (1985)]：即使对于一些小的初始数据，可压缩欧拉方程也不存在全局经典解。然后，我们将这种方法应用于二维空间维度的非线性磁流体动力学。最后，我们通过忽略三次项和高阶项来考虑由各向同性超弹性材料的弹性动力学引起的具有二次非线性的二阶拟线性双曲系统。