Dynamics and Singularities in Models of Inertial Transport of Mass

The talk presents a study of applications of nonlinear PDEs (the Hamilton-Jacobi equation, equations of pressureless gas dynamics) to mathematical cosmology and adjacent areas of mathematical physics. The main emphasis will be on the method of approximate reconstruction of large-scale component of the peculiar velocity field in the Universe from the observed spatial distribution of mass. The method is based on reconstruction of the gradient component of the displacement field. We discuss the mathematical foundations of the method, its algorithmic implementation, testing on synthetic catalogues of galaxies, and application to observational data that allowed to improve the existing estimates of the mean density of dark matter and the age of the Universe