On non-uniqueness for the system ${\text{bu}}_t+(\text{bu}\cdot \mathrm{\nabla })\text{bu}=\mu \mathrm{\Delta }\mathbf{u}$

Explicit irrotational solutions, obtained via the Cole-Hopf transform from the multi-d heat equation, give examples of non-uniqueness for the Cauchy problem in supercritical $L^p$, $W^{1,p}$, and $W^{2,p}$ regimes. We verify non-uniqueness of the trivial solution in the sense of $L^p({\text{RR}}^n)$, whenever $n\ge 2$ and $1\le p<n$. The same solutions give non-uniqueness in $W^{1,p}({\text{RR}}^n)$ and $W^{2,p}({\text{RR}}^n)$ for $1\le p<\frac{n}{2}$ and $1\le p<\frac{n}{3}$, respectively. The main example provides solutions which are classical for strictly positive times, and vanish in the stated norms, but explode in $L^{\mathrm{\infty }}({\text{RR}}^n)$, as $t\to 0+$. The non-uniqueness is unrelated to the Tikhonov non-uniqueness phenomenon for the heat equation.