Existence of regular solutions for a certain type of non-Newtonian Navier-Stokes equations

TL;DR

This paper proves the local and global existence of regular solutions for a specific class of non-Newtonian Navier-Stokes equations in three dimensions, depending on initial data smoothness and size.

Contribution

It establishes the local existence and uniqueness of regular solutions, and global existence under small initial data for certain non-Newtonian fluids.

Findings

Local existence of unique regular solutions

Global existence with small initial data

Solutions depend on initial data smoothness and size

Abstract

We are concerned with existence of regular solutions for non-Newtonian fluids in dimension three. For a certain type of non-Newtonian fluids we prove local existence of unique regular solutions, provided that the initial data are sufficiently smooth. Moreover, if the $H^3$-norm of initial data is sufficiently small, then the regular solution exists globally in time.