Fujita-Kato solution for the 3D compressible pressureless Navier-Stokes equations with discontinuous and large-variation density

This paper mainly focuses on the Cauchy problem to the 3D compressible pressureless Navier-Stokes equations arising from models of collective behavior, which can be derived by taking the high Mach number limit of the classical compressible Navier-Stokes system. We construct the global-in-time existence and uniqueness of the so-called Fujita-Kato solution to the system, provided that the initial density $ρ_0$ is discontinuous, large-variation and the initial velocity $u_0$ is in a critical functional framework. Our method relies on some time weighted estimates and the Lagrangian approach.