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

TL;DR

This paper establishes the global existence and uniqueness of Fujita-Kato solutions for 3D compressible pressureless Navier-Stokes equations with discontinuous, high-variation initial density, using time-weighted estimates and Lagrangian methods.

Contribution

It introduces a novel approach to prove global solutions for the pressureless Navier-Stokes system with discontinuous initial data.

Findings

Proved global-in-time existence of solutions.

Established uniqueness under critical initial conditions.

Handled discontinuous, large-variation initial densities.

Abstract

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 $\rho_0$ 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.