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

TL;DR

This paper constructs explicit solutions demonstrating non-uniqueness in the Cauchy problem for a Navier-Stokes type system in supercritical function spaces, showing solutions can vanish in certain norms yet explode in others.

Contribution

It provides explicit irrotational solutions via Cole-Hopf transform that prove non-uniqueness in supercritical $L^p$, $W^{1,p}$, and $W^{2,p}$ regimes for the Navier-Stokes system.

Findings

Non-uniqueness in $L^p( r^n)$ for $1 \\leq p < n$.

Non-uniqueness in $W^{1,p}$ for $1 \\leq p < n/2$.

Solutions can explode in $L^\\infty$ as $t \\to 0^+$.

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 $n\geq2$ 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.