# Wave breaking in a class of non-local conservation laws

## TL;DR

This paper generalizes Riccati-type systems to analyze wave breaking in non-local conservation laws, allowing for time-varying coefficients, and applies the theory to the Whitham-type equation with nonlinear drift.

## Contribution

It introduces a new Riccati-type system with time-varying coefficients for wave breaking analysis in non-local conservation laws, expanding previous fixed-coefficient models.

## Key findings

- Derived a generalized Riccati-type system for non-local flows.
- Proved wave breaking occurs under broader conditions with variable coefficients.
- Applied the theory to the Whitham-type equation demonstrating its effectiveness.

## Abstract

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.