On non-uniqueness for the system $\bu_t+(\bu\cdot\nabla)\bu=μΔ{\bf u}$

Abstract

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 $L^p(\RR^n)$, whenever $n\ge 2$ and $1\le p<n$. The same solutions give non-uniqueness in $W^{1,p}(\RR^n)$ and $W^{2,p}(\RR^n)$ 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 $L^\infty(\RR^n)$, as $t\to 0+$. The non-uniqueness is unrelated to the Tikhonov non-uniqueness phenomenon for the heat equation.