In 1757, Leonhard Euler wrote in his memoir Principes généreux du mouvement des fluides an equation for the conservation of momentum and another for the conservation of mass. These equations were among the first partial differential equations ever written and raised the initial problems that later led to the development of the domain of conservation laws with widespread applications in physics and chemistry. From a mathematical point of view, they are often classified as hyperbolic due to their wavelike solutions. Yet, they are famous for having shock singularities, requiring mostly an ad hoc mathematical framework and often being placed in the last chapter of PDE textbooks.

In this talk, I will review this mathematical framework. As an application, I will discuss in detail a system of two conservation laws that emerges as a scaling limit of an interacting particle system, namely the two-species totally asymmetric simple exclusion process (2-TASEP). This system is integrable at the microscopic level in the Yang-Baxter sense. Interestingly, its hydrodynamic limit PDEs are integrable in the sense that they belong to the Temple Class, making it a potential toy model to investigate the relation between the two notions of integrability.