Non Uniqueness of power-law flows

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 $L^2$.

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