The content of the talk is divided into two main parts. The first part discusses two non-standard mathematical models of continuous medium flow, which can be expressed as a quasi-linear system of conservation laws. A key feature of these systems is that, even under smooth initial conditions, their generalized solutions can have different types of singularities in the general case. First, we consider a one-dimensional system of equations for compressible two-phase multi-component filtration. It will be shown how concepts from the theory of conservation laws can be used to study this system, including, for example, the Riemann problem. However, solutions to this problem will not exhibit the typical characteristics of the standard theory. Instead, they will exhibit infinite propagation velocities and be always discontinuous. Second, we will consider dynamics in a two-dimensional isobaric medium, which is a system of equations of pressureless gas dynamics. It describes the phenomena of matter concentration. This system of equations leads to the emergence of strong singularities in the form of delta functions on manifolds of different dimensions. As a result, specific Rankine-Hugoniot-type equations arise. During the evolution process, singularities along curves in the plane interact with each other, leading to various configurations, including delta functions at a point. This process can be seen as the formation of an evolving hierarchy of singularities. In the second part of the talk, we will explore an alternative view on the nature of quasi-linear conservation laws systems based on a variational representation of generalized solutions. The form of this representation differs from traditional formulations used in the theory of second-order equations. Two such representations are discussed: 1) Based on the generalization of known results (starting with the works of E. Hopf), which is a variational representation of solutions for a single equation. 2) Based on the representation of generalized solutions as functionals in the trajectory space.
