In this work, we study constant-coefficient first order systems of partial differential equations and give necessary and sufficient conditions for those systems to have a well-posed Cauchy Problem. In many physical applications, due to the presence of constraints, the number of equations in the PDE system is larger than the number of unknowns, thus the standard Kreiss conditions can not be directly applied to check whether the system admits a well-posed initial value formulation. In this work, we find necessary and sufficient conditions such that there exists a reduced set of equations, of the same dimensionality as the set of unknowns, which satisfy Kreiss conditions and so are well defined and properly behaved evolution equations. We do that by studying the systems using the Kronecker decomposition of matrix pencils and, once the conditions are met, finding specific families of reductions which render the system strongly hyperbolic. We show the power of the theory in some examples: Klein Gordon, the ADM, and the BSSN equations by writing them as first order systems, and studying their Kronecker decomposition and general reductions.

中文翻译：

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带约束系统强双曲的充要条件

在这项工作中，我们研究了偏微分方程的常系数一阶系统，并给出了这些系统具有适定柯西问题的充分必要条件。在许多物理应用中，由于约束的存在，PDE系统中方程的个数大于未知数的个数，因此不能直接应用标准的Kreiss条件来检验系统是否接受适定初值公式。在这项工作中，我们找到了必要和充分条件，使得存在一组简化的方程，与一组未知数具有相同的维度，满足 Kreiss 条件，因此是定义明确且表现适当的进化方程。我们通过使用矩阵铅笔的 Kronecker 分解研究系统来做到这一点，一旦满足条件，找到特定的减少族，使系统具有强烈的双曲线性。我们在一些例子中展示了该理论的力量：Klein Gordon、ADM 和 BSSN 方程，将它们写成一阶系统，并研究它们的 Kronecker 分解和一般归约。