Existence of weak solutions to $p$-Navier-Stokes equations

TL;DR

This paper proves the existence of weak solutions to the $p$-Navier-Stokes equations on bounded domains, introducing a new basis construction and establishing energy dissipation properties.

Contribution

It constructs a divergence-free Schauder basis in $W_0^{1,p}( abla)$ and proves weak solution existence via Galerkin approximation, addressing gaps in previous work.

Findings

Existence of weak solutions for $p$-Navier-Stokes equations.

A new divergence-free Schauder basis in $W_0^{1,p}( abla)$.

Energy dissipation equality for weak solutions.

Abstract

We study the existence of weak solutions to the $p$-Navier-Stokes equations with a symmetric $p$-Laplacian on bounded domains. We construct a particular Schauder basis in $W_0^{1,p}(\Omega)$ with divergence free constraint and prove existence of weak solutions using the Galerkin approximation via this basis. Meanwhile, in the proof, we establish a chain rule for the $L^p$ norm of the weak solutions, which fixes a gap in our previous work. The equality of energy dissipation is also established for the weak solutions considered.