For models describing water waves, Constantin and Escher's works have long been considered as the cornerstone method for proving wave breaking phenomena. Their rigorous analytic proof shows that if the lowest slope of flows can be controlled by its highest slope initially, then the wave-breaking occur for the Whitham-type equation. Since this breakthrough, there have been numerous refined wave-breaking results established by generalizing the kernel which describes the dispersion relation of water waves. Even though the proofs of these involve a system of coupled Riccati-type differential inequalities, however, little or no attention has been made to a generalization of this Riccati-type system. In this work, from a rich class of non-local conservation laws, a Riccati-type system that governs the flow's gradient is extracted and investigated. The system's leading coefficient functions are allowed to change their values and signs over time as opposed to the ones in many of other previous works are fixed constants. Up to the author's knowledge, the blow-up analysis upon this structural generalization is new and is of theoretical interest in itself as well as its application to various non-local flow models. The theory is illustrated via the Whitham-type equation with nonlinear drift. Our method is applicable to a large class of non-local conservation laws.

中文翻译：

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一类非局域守恒定律中的波浪破碎

对于描述水波的模型，康斯坦丁和埃舍尔的作品长期以来一直被认为是证明破浪现象的基石方法。他们严格的分析证明表明，如果流动的最低坡度最初可以由其最高坡度控制，那么 Whitham 型方程就会发生波浪破碎。自从这一突破以来，通过推广描述水波色散关系的核，已经建立了许多精细的波浪破碎结果。尽管这些证明涉及耦合 Riccati 型微分不等式的系统，但是，很少或根本没有注意到对这个 Riccati 型系统的推广。在这项工作中，从丰富的非局部守恒定律中，一个控制流动的 Riccati 型系统' s 梯度被提取和研究。系统的主要系数函数可以随时间改变它们的值和符号，而不是许多其他以前的作品中的那些是固定常数。据作者所知，对这种结构概括的全面分析是新的，其本身具有理论意义，也适用于各种非局部流动模型。该理论通过具有非线性漂移的 Whitham 型方程来说明。我们的方法适用于一大类非局部守恒定律。对这种结构概括的全面分析是新的，它本身具有理论意义，也适用于各种非局部流动模型。该理论通过具有非线性漂移的 Whitham 型方程来说明。我们的方法适用于一大类非局部守恒定律。对这种结构概括的全面分析是新的，它本身具有理论意义，也适用于各种非局部流动模型。该理论通过具有非线性漂移的 Whitham 型方程来说明。我们的方法适用于一大类非局部守恒定律。