Development of singular surfaces and formation of generic singularities in 3D pressureless flows

In this paper, the Cauchy problem for the equations of pressureless flows in three space dimensions is considered. It is well known that for smooth initial data, the density and velocity gradient often become unbounded at a finite time $t_0$. This type of singularity is associated with sticky-particle dynamics. For an open dense set of smooth initial data, the formation and structure of singularities are investigated using the transversality theorem, with the aid of constructive methods for initial velocity fields. In contrast to the two-dimensional case studied by Bressan, Chen and Huang (arXiv:2408.06877), for $t>t_0$ a singular hypersurface will be formed, whose governing system of equations is not hyperbolic. By assuming the initial data are analytic, we are able to precisely analyze the singular behavior of velocity and density. Based on this, the system of equations for singular hypersurfaces can be recast into a first-order system of Briot-Bouquet type, and the local analytic solution can be constructed using a version of the Cauchy-Kovalevskaya theorem.