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 $n\geq2$ and $1\leq p<n$. The same solutions give non-uniqueness in $W^{ 1,p}(\RR^n)$ and $W^{ 2,p}(\RR^n)$ for $1\leq p<\frac{ n}{ 2}$ and $1\leq p<\frac{ n}{ 3}$, 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 $t\to0+$. The non-uniqueness is unrelated to the Tikhonov non-uniqueness phenomenon for the heat equation.
Categories: math.AP