Non Uniqueness of power-law flows

TL;DR

This paper demonstrates non-uniqueness and ill-posedness of solutions for power-law fluids in multiple dimensions using convex integration, extending previous results to broader ranges of the power index and solution classes.

Contribution

It introduces convex integration techniques to establish non-uniqueness and ill-posedness for power-law fluids across wider parameter ranges and solution types.

Findings

Ill-posedness for $q otin (1, rac{2d}{d+2})$ in Leray-Hopf solutions.

Ill-posedness for $q otin (1, rac{3d+2}{d+2})$ in distributional solutions.

Construction of non-unique solutions for all data in $L^2$.

Abstract

We apply the technique of convex integration to obtain non-uniqueness and existence results for power-law fluids, in dimension $d\ge 2$. For the power index $q$ below the compactness threshold, i.e. $q \in (1, \frac{2d}{d+2})$, we show ill-posedness of Leray-Hopf solutions. For a wider class of indices $q \in (1, \frac{3d+2}{d+2})$ we show ill-posedness of distributional (non-Leray-Hopf) solutions, extending the seminal paper of Buckmaster and Vicol. In this wider class we also construct non-unique solutions for every datum in $L^2$.