Local well-posedness and singularity formation in non-Newtonian compressible fluids

TL;DR

This paper establishes local well-posedness for a broad class of non-Newtonian compressible fluids and demonstrates that singularities can form in finite time, indicating potential breakdown of solutions.

Contribution

It proves local well-posedness and shows finite-time singularity formation for non-Newtonian compressible fluids, a novel result in the field.

Findings

Local well-posedness of the initial value problem

Existence of initial data leading to finite-time singularities

Breakdown of solutions in finite time for certain initial conditions

Abstract

We investigate the initial value problem of a very general class of $3+1$ non-Newtonian compressible fluids in which the viscous stress tensor with shear and bulk viscosity relaxes to its Navier-Stokes values. These fluids correspond to the non-relativistic limit of well-known Israel-Stewart-like theories used in the relativistic fluid dynamic simulations of high-energy nuclear and astrophysical systems. After establishing the local well-posedness of the Cauchy problem, we show for the first time in the literature that there exists a large class of initial data for which the corresponding evolution breaks down in finite time due to the formation of singularities. This implies that a large class of non-Newtonian fluids do not have finite solutions defined at all times.